____________________________________________________________________________________
Added March 10, 2019.
Problem Chapter 5.2.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c w + \beta x y+\gamma \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x, y] + beta*x*y + gamma; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {-2 a b \beta +c^3 e^{\frac {c x}{a}} c_1\left (\frac {a y-b x}{a}\right )-a \beta c y-b \beta c x-\beta c^2 x y-c^2 \gamma }{c^3}\right \}\right \} \]
Maple ✓
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+beta*x*y+gamma; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) = \left ( -{\frac {\beta \,y}{c}}-{\frac {b\beta }{{c}^{2}}} \right ) x-{\frac {ya\beta }{{c}^{2}}}+{\frac {1}{{c}^{3}} \left ( {\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {{\rm e}^{{\frac {cx}{a}}}}{c}^{3}-\gamma \,{c}^{2}-2\,ab\beta \right ) } \]
____________________________________________________________________________________
Added March 10, 2019.
Problem Chapter 5.2.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c w + x(\beta x+\gamma y)+\delta \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x, y] + x*(beta*x + gamma*y) + delta; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {-2 a^2 \beta +c^3 e^{\frac {c x}{a}} c_1\left (\frac {a y-b x}{a}\right )-2 a b \gamma -2 a \beta c x-a c \gamma y-b c \gamma x-\beta c^2 x^2-c^2 \delta -c^2 \gamma x y}{c^3}\right \}\right \} \]
Maple ✓
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+x*(beta*x+gamma*y)+delta; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) =-{\frac {\beta \,{x}^{2}}{c}}+ \left ( -{\frac {\gamma \,y}{c}}+{\frac {-2\,\beta \,ac-bc\gamma }{{c}^{3}}} \right ) x-{\frac {\gamma \,ya}{{c}^{2}}}+{\frac {1}{{c}^{3}} \left ( {\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {{\rm e}^{{\frac {cx}{a}}}}{c}^{3}-2\,a\gamma \,b-2\,{a}^{2}\beta -\delta \,{c}^{2} \right ) } \]
____________________________________________________________________________________
Added March 10, 2019.
Problem Chapter 5.2.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + y w_y = w + a x^2+b y^2+c \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0]; pde = x*D[w[x, y], x] + y*D[w[x, y], y] == w[x, y] + a*x^2 + b*y^2 + c; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to a x^2+b y^2+x c_1\left (\frac {y}{x}\right )-c\right \}\right \} \]
Maple ✓
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); pde := x*diff(w(x,y),x)+ y*diff(w(x,y),y) = w(x,y)+a*x^2+b*y^2+c; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) =b{y}^{2}+a{x}^{2}+{\it \_F1} \left ( {\frac {y}{x}} \right ) x-c \]
____________________________________________________________________________________
Added March 10, 2019.
Problem Chapter 5.2.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x w_x + b y w_y = c w + x(\beta x+\gamma y)+ \delta \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0]; pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == c*w[x, y] + x*(beta*x + gamma*y) + delta; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {2 a^2 c x^{\frac {c}{a}} c_1\left (y x^{-\frac {b}{a}}\right )-2 a^2 \delta +c^3 x^{\frac {c}{a}} c_1\left (y x^{-\frac {b}{a}}\right )-3 a c^2 x^{\frac {c}{a}} c_1\left (y x^{-\frac {b}{a}}\right )-b c^2 x^{\frac {c}{a}} c_1\left (y x^{-\frac {b}{a}}\right )+2 a b c x^{\frac {c}{a}} c_1\left (y x^{-\frac {b}{a}}\right )-2 a b \delta +a \beta c x^2+3 a c \delta +2 a c \gamma x y+b \beta c x^2+b c \delta -\beta c^2 x^2-c^2 \delta -c^2 \gamma x y}{c (c-2 a) (-a-b+c)}\right \}\right \} \]
Maple ✓
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); pde := a*x*diff(w(x,y),x)+ b*y*diff(w(x,y),y) = c*w(x,y)+x*(beta*x+gamma*y)+delta; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\frac {\gamma \,y}{a}{x}^{{\frac {a+b}{a}}-{\frac {b}{a}}} \left ( {\frac {b-c}{a}}+1 \right ) ^{-1}}+{\frac {\beta \,{x}^{2}}{a} \left ( {\frac {a-c}{a}}+1 \right ) ^{-1}}+{\frac {\delta }{a} \left ( 1-{\frac {a+c}{a}} \right ) ^{-1}}+{x}^{{\frac {c}{a}}}{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) \]
____________________________________________________________________________________
Added March 10, 2019.
Problem Chapter 5.2.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a y w_x + (b_2 x^2+b_1 x+b_0) w_y = (c_2 x^2+c_1 x+c_0) w + s_2 x^2+s_1 x+s_0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0]; pde = a*y*D[w[x, y], x] + (b2*x^2 + b1*x + b0)*D[w[x, y], y] == (c2*x^2 + c1*x + c0)*w[x, y] + s2*x^2 + s1*x + s0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {\$Aborted} \] Timed out
Maple ✓
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); pde := a*y*diff(w(x,y),x)+ (b2*x^2+b1*x+b0)*diff(w(x,y),y) = (c2*x^2+c1*x+c0)*w(x,y)+s2*x^2+s1*x+s0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); sol:=simplify(sol,size);
\[ \text {Too large to display} \]
____________________________________________________________________________________
Added March 10, 2019.
Problem Chapter 5.2.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a y^2 w_x + (b_1 x^2+b_0) w_y = (c_1 x^2+c_0) w + s_1 x^2+s_0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0]; pde = a*y^2*D[w[x, y], x] + (b1*x^2 + b0)*D[w[x, y], y] == (c1*x^2 + c0)*w[x, y] + s1*x^2 + s0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {\$Aborted} \] Timed out
Maple ✓
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); pde := a*y*diff(w(x,y),x)+ (b1*x^2+b0)*diff(w(x,y),y) = (c1*x^2+c0)*w(x,y)+s1*x^2+s0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) =- \left ( \sqrt {3}\int ^{x}\!{\frac {{{\it \_b}}^{2}{\it s1}+{\it s0}}{\sqrt {a \left ( 2\,{{\it \_b}}^{3}{\it b1}-2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a+6\,{\it \_b}\,{\it b0}-6\,{\it b0}\,x \right ) }}{{\rm e}^{\sqrt {3} \left ( {\frac {2/3\,i{\it c0}\,\sqrt {3}}{\sqrt {2\,{{\it \_b}}^{3}a{\it b1}+a \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) +6\,{\it \_b}\,a{\it b0}}} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \sqrt {{-i\sqrt {3} \left ( {\it \_b}+1/4\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}-{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}}+i/2\sqrt {3} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \right ) \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) ^{-1}}}\sqrt {{1 \left ( {\it \_b}-1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \left ( -3/4\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+3\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}}-i/2\sqrt {3} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \right ) ^{-1}}}\sqrt {{i\sqrt {3} \left ( {\it \_b}+1/4\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}-{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}}-i/2\sqrt {3} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \right ) \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) ^{-1}}}\EllipticF \left ( 1/3\,\sqrt {3}\sqrt {{-i\sqrt {3} \left ( {\it \_b}+1/4\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}-{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}}+i/2\sqrt {3} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \right ) \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) ^{-1}}},\sqrt {{-i\sqrt {3} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \left ( -3/4\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+3\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}}-i/2\sqrt {3} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \right ) ^{-1}}} \right ) }+{\it c1}\, \left ( 1/3\,{\frac {\sqrt {2\,{{\it \_b}}^{3}a{\it b1}+a \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) +6\,{\it \_b}\,a{\it b0}}}{a{\it b1}}}-{\frac {2/3\,i{\it b0}\,\sqrt {3}}{{\it b1}\,\sqrt {2\,{{\it \_b}}^{3}a{\it b1}+a \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) +6\,{\it \_b}\,a{\it b0}}} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \sqrt {{-i\sqrt {3} \left ( {\it \_b}+1/4\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}-{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}}+i/2\sqrt {3} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \right ) \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) ^{-1}}}\sqrt {{1 \left ( {\it \_b}-1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \left ( -3/4\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+3\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}}-i/2\sqrt {3} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \right ) ^{-1}}}\sqrt {{i\sqrt {3} \left ( {\it \_b}+1/4\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}-{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}}-i/2\sqrt {3} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \right ) \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) ^{-1}}}\EllipticF \left ( 1/3\,\sqrt {3}\sqrt {{-i\sqrt {3} \left ( {\it \_b}+1/4\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}-{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}}+i/2\sqrt {3} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \right ) \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) ^{-1}}},\sqrt {{-i\sqrt {3} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \left ( -3/4\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+3\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}}-i/2\sqrt {3} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \right ) ^{-1}}} \right ) } \right ) \right ) }}}{d{\it \_b}}-{\it \_F1} \left ( 1/3\,{\frac {-2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x}{a}} \right ) \right ) {{\rm e}^{\sqrt {3} \left ( {\frac {-2/3\,i{\it c0}\,\sqrt {3}}{\sqrt {2\,a{\it b1}\,{x}^{3}+a \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) +6\,a{\it b0}\,x}} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \sqrt {{-i\sqrt {3} \left ( x+1/4\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}-{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}}+i/2\sqrt {3} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \right ) \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) ^{-1}}}\sqrt {{1 \left ( x-1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \left ( -3/4\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+3\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}}-i/2\sqrt {3} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \right ) ^{-1}}}\sqrt {{i\sqrt {3} \left ( x+1/4\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}-{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}}-i/2\sqrt {3} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \right ) \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) ^{-1}}}\EllipticF \left ( 1/3\,\sqrt {3}\sqrt {{-i\sqrt {3} \left ( x+1/4\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}-{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}}+i/2\sqrt {3} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \right ) \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) ^{-1}}},\sqrt {{-i\sqrt {3} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \left ( -3/4\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+3\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}}-i/2\sqrt {3} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \right ) ^{-1}}} \right ) }-{\it c1}\, \left ( 1/3\,{\frac {\sqrt {2\,a{\it b1}\,{x}^{3}+a \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) +6\,a{\it b0}\,x}}{a{\it b1}}}-{\frac {2/3\,i{\it b0}\,\sqrt {3}}{{\it b1}\,\sqrt {2\,a{\it b1}\,{x}^{3}+a \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) +6\,a{\it b0}\,x}} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \sqrt {{-i\sqrt {3} \left ( x+1/4\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}-{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}}+i/2\sqrt {3} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \right ) \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) ^{-1}}}\sqrt {{1 \left ( x-1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \left ( -3/4\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+3\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}}-i/2\sqrt {3} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \right ) ^{-1}}}\sqrt {{i\sqrt {3} \left ( x+1/4\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}-{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}}-i/2\sqrt {3} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \right ) \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) ^{-1}}}\EllipticF \left ( 1/3\,\sqrt {3}\sqrt {{-i\sqrt {3} \left ( x+1/4\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}-{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}}+i/2\sqrt {3} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \right ) \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) ^{-1}}},\sqrt {{-i\sqrt {3} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \left ( -3/4\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+3\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}}-i/2\sqrt {3} \left ( 1/2\,{\frac {1}{{\it b1}}\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}+2\,{{\it b0}{\frac {1}{\sqrt [3]{ \left ( 4\,{\it b1}\,{x}^{3}-6\,{y}^{2}a+12\,{\it b0}\,x+2\,\sqrt {{\frac { \left ( -2\,{\it b1}\,{x}^{3}+3\,{y}^{2}a-6\,{\it b0}\,x \right ) ^{2}{\it b1}+16\,{{\it b0}}^{3}}{{\it b1}}}} \right ) {{\it b1}}^{2}}}}} \right ) \right ) ^{-1}}} \right ) } \right ) \right ) }} \]
____________________________________________________________________________________
Added March 10, 2019.
Problem Chapter 5.2.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a_1 x^2+a_0) w_x + (y+b_2 x^2+b_1 x+b_0) w_y = (c_2 y+c_1 x+c_0) w + k_{22}y^2+k{12} x y+k_{11} x^2+k_0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12]; pde = (a1*x^2 + a0)*y^2*D[w[x, y], x] + (y + b2*x^2 + b1*x + b0)*D[w[x, y], y] == (c2*y + c1*x + c0)*w[x, y] + k22*y^2 + k12*x*y + k11*x^2 + k0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); pde := (a1*x^2+a0)*diff(w(x,y),x)+ (y+b2*x^2+b1*x+b0)*diff(w(x,y),y) = (c2*y+c1*x+c0)*w(x,y)+ k22*y^2+k12*x*y+k11*x^2+k0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{{{\it \_f}}^{2}{\it a1}+{\it a0}} \left ( \left ( \int \!{\frac {{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_f} \right ) ^{2}{\it k22}\,{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}} \left ( \int \!{\frac {1}{{{\it \_f}}^{2}{\it a1}+{\it a0}} \left ( {\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}\int \!{\frac {{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_f}+{\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}} \left ( -\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}} \right ) +{\it c1}\,{\it \_f}+{\it c0} \right ) }\,{\rm d}{\it \_f}\sqrt {{\it a0}\,{\it a1}}-2\,\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) \right ) }}}+2\,\int \!{\frac {{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_f} \left ( -\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}} \right ) {\it k22}\,{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}} \left ( \int \!{\frac {1}{{{\it \_f}}^{2}{\it a1}+{\it a0}} \left ( {\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}\int \!{\frac {{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_f}+{\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}} \left ( -\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}} \right ) +{\it c1}\,{\it \_f}+{\it c0} \right ) }\,{\rm d}{\it \_f}\sqrt {{\it a0}\,{\it a1}}-2\,\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) \right ) }}}+{\it k12}\,{\it \_f}\,\int \!{\frac {{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_f}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}} \left ( \int \!{\frac {1}{{{\it \_f}}^{2}{\it a1}+{\it a0}} \left ( {\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}\int \!{\frac {{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_f}+{\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}} \left ( -\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}} \right ) +{\it c1}\,{\it \_f}+{\it c0} \right ) }\,{\rm d}{\it \_f}\sqrt {{\it a0}\,{\it a1}}-\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) \right ) }}}+ \left ( -\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}} \right ) ^{2}{\it k22}\,{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}} \left ( \int \!{\frac {1}{{{\it \_f}}^{2}{\it a1}+{\it a0}} \left ( {\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}\int \!{\frac {{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_f}+{\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}} \left ( -\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}} \right ) +{\it c1}\,{\it \_f}+{\it c0} \right ) }\,{\rm d}{\it \_f}\sqrt {{\it a0}\,{\it a1}}-2\,\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) \right ) }}}+{\it k12}\,{\it \_f}\, \left ( -\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}} \right ) {{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}} \left ( \int \!{\frac {1}{{{\it \_f}}^{2}{\it a1}+{\it a0}} \left ( {\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}\int \!{\frac {{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_f}+{\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}} \left ( -\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}} \right ) +{\it c1}\,{\it \_f}+{\it c0} \right ) }\,{\rm d}{\it \_f}\sqrt {{\it a0}\,{\it a1}}-\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) \right ) }}}+{{\rm e}^{-\int \!{\frac {1}{{{\it \_f}}^{2}{\it a1}+{\it a0}} \left ( {\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}\int \!{\frac {{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_f}+{\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}} \left ( -\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}} \right ) +{\it c1}\,{\it \_f}+{\it c0} \right ) }\,{\rm d}{\it \_f}}}{\it k11}\,{{\it \_f}}^{2}+{{\rm e}^{-\int \!{\frac {1}{{{\it \_f}}^{2}{\it a1}+{\it a0}} \left ( {\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}\int \!{\frac {{{\it \_f}}^{2}{\it b2}+{\it \_f}\,{\it b1}+{\it b0}}{{{\it \_f}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_f}+{\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_f}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}} \left ( -\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}} \right ) +{\it c1}\,{\it \_f}+{\it c0} \right ) }\,{\rm d}{\it \_f}}}{\it k0} \right ) }{d{\it \_f}}+{\it \_F1} \left ( -\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{{{\it \_b}}^{2}{\it a1}+{\it a0}} \left ( {\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_b}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}\int \!{\frac {{{\it \_b}}^{2}{\it b2}+{\it \_b}\,{\it b1}+{\it b0}}{{{\it \_b}}^{2}{\it a1}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_b}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}{\it \_b}+{\it c2}\,{{\rm e}^{{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,{\it \_b}}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}} \left ( -\int \!{\frac {{\it b2}\,{x}^{2}+{\it b1}\,x+{\it b0}}{{\it a1}\,{x}^{2}+{\it a0}}{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}}}\,{\rm d}x+y{{\rm e}^{-{\frac {1}{\sqrt {{\it a0}\,{\it a1}}}\arctan \left ( {\frac {{\it a1}\,x}{\sqrt {{\it a0}\,{\it a1}}}} \right ) }}} \right ) +{\it c1}\,{\it \_b}+{\it c0} \right ) }{d{\it \_b}}}} \]
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Added March 10, 2019.
Problem Chapter 5.2.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a_1 x^2+a_0) w_x + (b_2 y^2+b_1 x y) w_y = (c_2 y^2+c_1 x^2) w + s_{22}y^2+s_{12} x y+s_{11} x^2+s_0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12]; pde = (a1*x^2 + a0)*y^2*D[w[x, y], x] + (b2*y^2 + b1*x^2)*D[w[x, y], y] == (c2*y^2 + c1*x^2)*w[x, y] + s22*y^2 + s12*x*y + s11*x^2 + s0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); pde := (a1*x^2+a0)*diff(w(x,y),x)+ (b2*y^2+b1*x^2)*diff(w(x,y),y) = (c2*y^2+c1*x^2)*w(x,y)+ s22*y^2+s12*x*y+s11*x^2+s0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); sol:=simplify(sol,size);
\[ \text {Too large to display} \]