____________________________________________________________________________________
Added April 15, 2019.
Problem Chapter 6.2.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ 2 b^2 x z w_x + b y (b^2 z^2 +1 ) w_y + a x y (b z +1)^2 w_z= 0 \]
Mathematica ✗
ClearAll[w, x, y, z, 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,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde =2*b^2*x*z*D[w[x, y,z], x] + b*y*(b^2*z^2 +1)*D[w[x, y,z], y] + a*x*y*(b*z +1)^2*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \text {Failed} \]
Maple ✗
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,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,sigma,lambda1,lambda2,n1,n2,nu'); pde := 2*b^2*x*z*diff(w(x,y,z),x)+b*y*(b^2*z^2 +1)*diff(w(x,y,z),y)+a*x*y*(b*z +1)^2*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ \text { sol=() } \]
____________________________________________________________________________________
Added April 15, 2019.
Problem Chapter 6.2.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ b c x y^2 w_x + 2 b c y^3 w_y + 2(c y z-a x^2)^2 w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, 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,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde =b*c*x*y^2*D[w[x, y,z], x] + 2*b*c*y^3*D[w[x, y,z], y] + 2*(c*y*z-a*x^2)^2*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (\frac {y}{x^2},\frac {x^4 \left (\log (x) \left (2 c y z-2 a x^2\right )+b y\right )}{b c y^2 \left (a x^2-c y z\right )}\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,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,sigma,lambda1,lambda2,n1,n2,nu'); pde := b*c*x*y^2*diff(w(x,y,z),x)+2*b*c*y^3*diff(w(x,y,z),y)+2*(c*y*z-a*x^2)^2*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {y}{{x}^{2}}},{\frac { \left ( -2\,a{x}^{2}+2\,cyz \right ) \ln \left ( x \right ) +by}{2\,a{x}^{2}-2\,cyz}} \right ) \]
____________________________________________________________________________________
Added April 15, 2019.
Problem Chapter 6.2.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ b c^2 y^2 z w_x + a c^2 x z^2 w_y - a b x y^2 w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, 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,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = b*c*x*y^2*D[w[x, y,z], x] + a*c^2*x*z^2*D[w[x, y,z], y] - a*b*x*y^2*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (\frac {a x}{c}+z,\frac {b y^3+c^2 z^3}{3 b}\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,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,sigma,lambda1,lambda2,n1,n2,nu'); pde := b*c*x*y^2*diff(w(x,y,z),x)+a*c^2*x*z^2*diff(w(x,y,z),y)- a*b*x*y^2*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ax+cz}{c}},{\frac {-{a}^{3}{x}^{3}-3\,{a}^{2}c{x}^{2}z-3\,a{c}^{2}x{z}^{2}+bc{y}^{3}}{bc}} \right ) \]
____________________________________________________________________________________
Added April 15, 2019.
Problem Chapter 6.2.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x(b y^2-c z^2) w_x + y(c z^2-a x^2) w_y + z(a x^2-b y^2) w_z= 0 \]
Mathematica ✗
ClearAll[w, x, y, z, 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,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = x*(b*y^2-c*z^2)*D[w[x, y,z], x] + y*(c*z^2-a*x^2)*D[w[x, y,z], y] + z*(a*x^2-b*y^2)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \text {Failed} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,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,sigma,lambda1,lambda2,n1,n2,nu'); pde :=x*(b*y^2-c*z^2)*diff(w(x,y,z),x)+ y*(c*z^2-a*x^2)*diff(w(x,y,z),y) + z*(a*x^2-b*y^2)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={{\rm e}^{-1/4\,{\it \_C1}\,{x}^{2}}}{{\rm e}^{{\it \_C2}/4}}{{\rm e}^{-1/4\,{\frac {{\it \_C1}\,b{y}^{2}}{a}}}}{x}^{{\it \_C2}/2}{\it \_C3}\,{\it \_C5}\,{y}^{{\it \_C2}/2}{\it \_C4}\,{z}^{{\it \_C2}/2}{{\rm e}^{-1/4\,{\frac {{\it \_C1}\,c{z}^{2}}{a}}}} \]
____________________________________________________________________________________
Added April 15, 2019.
Problem Chapter 6.2.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ b y (3 a x^2+ b y^2+c z^2) w_x - 2 a x(a x^2+c z^2) w_y + 2 a b x y z w_z= 0 \]
Mathematica ✗
ClearAll[w, x, y, z, 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,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = b*y*(3*a*x^2+ b*y^2+c*z^2)*D[w[x, y,z], x] - 2*a*x*(a*x^2+c*z^2)*D[w[x, y,z], y] + 2*a*b*x*y*z*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \text {Failed} \]
Maple ✗
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,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,sigma,lambda1,lambda2,n1,n2,nu'); pde := b*y*(3*a*x^2+ b*y^2+c*z^2)*diff(w(x,y,z),x)- 2*a*x*(a*x^2+c*z^2)*diff(w(x,y,z),y) + 2*a*b*x*y*z*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ \text { sol=() } \]
____________________________________________________________________________________
Added April 15, 2019.
Problem Chapter 6.2.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ b(a(a^2 x^2+b^2 y^2-1) x+ b y ) w_x + a(b(a^2 x^2+b^2 y^2-1)y - a x) w_y + 2 a b z w_z= 0 \]
Mathematica ✗
ClearAll[w, x, y, z, 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,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = b*(a*(a^2*x^2+b^2*y^2-1)*x+ b*y )*D[w[x, y,z], x] +a*(b*(a^2*x^2+b^2*y^2-1)*y - a*x)*D[w[x, y,z], y] + 2*a*b*z*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \text {Failed} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,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,sigma,lambda1,lambda2,n1,n2,nu'); pde := b*(a*(a^2*x^2+b^2*y^2-1)*x+ b*y )*diff(w(x,y,z),x)+a*(b*(a^2*x^2+b^2*y^2-1)*y - a*x)*diff(w(x,y,z),y) + 2*a*b*z*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {{a}^{2}{x}^{2}+{y}^{2}{b}^{2}-1}{{a}^{2}{x}^{2}+{y}^{2}{b}^{2}}{{\rm e}^{2\,\arctan \left ( {\frac {by}{ax}} \right ) }}},z{{\rm e}^{-2\,\int ^{x}\!{\frac {1}{{\it \_a}} \left ( \cos \left ( \RootOf \left ( 2\,{\it \_Z}-\ln \left ( -{\frac {{{\it \_a}}^{2} \left ( {a}^{2}{x}^{2}+{y}^{2}{b}^{2}-1 \right ) {a}^{2}}{ \left ( {a}^{2}{x}^{2}+{y}^{2}{b}^{2} \right ) \left ( -{{\it \_a}}^{2}{a}^{2}+ \left ( \cos \left ( {\it \_Z} \right ) \right ) ^{2} \right ) }{{\rm e}^{2\,\arctan \left ( {\frac {by}{ax}} \right ) }}} \right ) \right ) \right ) \right ) ^{2} \left ( {{\it \_a}}^{2}{a}^{2}+\sin \left ( \RootOf \left ( 2\,{\it \_Z}-\ln \left ( -{\frac {{{\it \_a}}^{2} \left ( {a}^{2}{x}^{2}+{y}^{2}{b}^{2}-1 \right ) {a}^{2}}{ \left ( {a}^{2}{x}^{2}+{y}^{2}{b}^{2} \right ) \left ( -{{\it \_a}}^{2}{a}^{2}+ \left ( \cos \left ( {\it \_Z} \right ) \right ) ^{2} \right ) }{{\rm e}^{2\,\arctan \left ( {\frac {by}{ax}} \right ) }}} \right ) \right ) \right ) \cos \left ( \RootOf \left ( 2\,{\it \_Z}-\ln \left ( -{\frac {{{\it \_a}}^{2} \left ( {a}^{2}{x}^{2}+{y}^{2}{b}^{2}-1 \right ) {a}^{2}}{ \left ( {a}^{2}{x}^{2}+{y}^{2}{b}^{2} \right ) \left ( -{{\it \_a}}^{2}{a}^{2}+ \left ( \cos \left ( {\it \_Z} \right ) \right ) ^{2} \right ) }{{\rm e}^{2\,\arctan \left ( {\frac {by}{ax}} \right ) }}} \right ) \right ) \right ) - \left ( \cos \left ( \RootOf \left ( 2\,{\it \_Z}-\ln \left ( -{\frac {{{\it \_a}}^{2} \left ( {a}^{2}{x}^{2}+{y}^{2}{b}^{2}-1 \right ) {a}^{2}}{ \left ( {a}^{2}{x}^{2}+{y}^{2}{b}^{2} \right ) \left ( -{{\it \_a}}^{2}{a}^{2}+ \left ( \cos \left ( {\it \_Z} \right ) \right ) ^{2} \right ) }{{\rm e}^{2\,\arctan \left ( {\frac {by}{ax}} \right ) }}} \right ) \right ) \right ) \right ) ^{2} \right ) ^{-1}}{d{\it \_a}}}} \right ) \]
____________________________________________________________________________________
Added April 15, 2019.
Problem Chapter 6.2.3.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x(b^3 y^3 - 2 a^3 x^3) w_x + y(2 b^3 y^3 -a^3 x^3) w_y + 9 z(a^3 x^3-b^3 y^3) w_z= 0 \]
Mathematica ✗
ClearAll[w, x, y, z, 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,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = x*(b^3*y^3 - 2*a^3*x^3)*D[w[x, y,z], x] +y*(2*b^3*y^3 -a^3*x^3)*D[w[x, y,z], y] +9*z*(a^3*x^3-b^3*y^3)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \text {\$Aborted} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,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,sigma,lambda1,lambda2,n1,n2,nu'); pde := x*(b^3*y^3 - 2*a^3*x^3)*diff(w(x,y,z),x)+y*(2*b^3*y^3 -a^3*x^3)*diff(w(x,y,z),y) + 9*z*(a^3*x^3-b^3*y^3)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) ,z{{\rm e}^{9\,\int ^{x}\!{\frac {1}{972\,{\it \_a}} \left ( -9\, \left ( {a}^{3}{b}^{6}-1/9\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,a{{\it \_a}}^{3} \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) \right ) ^{3}}{b}^{3}-2/9\, \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) \right ) ^{3}{{\it \_a}}^{3} \right ) \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) {\it \_a}\, \left ( -108\,{a}^{3}{b}^{6}+8\, \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) \right ) ^{3}{{\it \_a}}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,a{{\it \_a}}^{3} \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) \right ) ^{3}}{b}^{3} \right ) ^{2/3}-36\, \left ( {a}^{3}{b}^{6}-1/9\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,a{{\it \_a}}^{3} \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) \right ) ^{3}}{b}^{3}-1/9\, \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) \right ) ^{3}{{\it \_a}}^{3} \right ) \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) \right ) ^{2}{{\it \_a}}^{2}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\, \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) \right ) ^{3}{{\it \_a}}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,a{{\it \_a}}^{3} \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) \right ) ^{3}}{b}^{3}}+648\, \left ( {a}^{3}{b}^{6}-1/9\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,a{{\it \_a}}^{3} \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) \right ) ^{3}}{b}^{3}-{\frac {2\, \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) \right ) ^{3}{{\it \_a}}^{3}}{27}} \right ) \left ( {a}^{3}{b}^{6}-1/6\, \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) \right ) ^{3}{{\it \_a}}^{3} \right ) \right ) \left ( -{\frac { \left ( {a}^{3}{b}^{6}-1/9\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,a{{\it \_a}}^{3} \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) \right ) ^{3}}{b}^{3}-2/9\, \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) \right ) ^{3}{{\it \_a}}^{3} \right ) \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) {\it \_a}\, \left ( -108\,{a}^{3}{b}^{6}+8\, \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) \right ) ^{3}{{\it \_a}}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,a{{\it \_a}}^{3} \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) \right ) ^{3}}{b}^{3} \right ) ^{2/3}}{108}}-1/27\, \left ( {a}^{3}{b}^{6}-1/9\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,a{{\it \_a}}^{3} \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) \right ) ^{3}}{b}^{3}-1/9\, \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) \right ) ^{3}{{\it \_a}}^{3} \right ) \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) \right ) ^{2}{{\it \_a}}^{2}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\, \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) \right ) ^{3}{{\it \_a}}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,a{{\it \_a}}^{3} \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) \right ) ^{3}}{b}^{3}}+ \left ( {a}^{3}{b}^{6}-1/9\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,a{{\it \_a}}^{3} \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) \right ) ^{3}}{b}^{3}-{\frac {2\, \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) \right ) ^{3}{{\it \_a}}^{3}}{27}} \right ) \left ( {a}^{3}{b}^{6}-1/9\, \left ( \RootOf \left ( -6\,y{b}^{3}\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}+4\,{x}^{3}{{\it \_Z}}^{2}+2\,\sqrt [3]{-108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3}}{x}^{2}{\it \_Z}+ \left ( -108\,{a}^{3}{b}^{6}+8\,{{\it \_Z}}^{3}{x}^{3}+12\,a\sqrt {3}\sqrt {27\,{a}^{4}{b}^{6}-4\,{{\it \_Z}}^{3}a{x}^{3}}{b}^{3} \right ) ^{2/3}x \right ) \right ) ^{3}{{\it \_a}}^{3} \right ) \right ) ^{-1}}{d{\it \_a}}}} \right ) \]
____________________________________________________________________________________
Added April 15, 2019.
Problem Chapter 6.2.3.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x^2(a b x y-c^2 z^2) w_x + a x y(a b x y-c^2 z^2) w_y + b y z(b c y z+2 a^2 x^2) w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, 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,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = a*x^2*(a*b*x*y-c^2*z^2)*D[w[x, y,z], x] +a*x*y*(a*b*x*y-c^2*z^2)*D[w[x, y,z], y] +b*y*z*(b*c*y*z+2*a^2*x^2) *D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (\frac {y}{x},\frac {\log \left (\frac {x z}{a^2 b x^2 y+a c^2 x z^2+b^2 c y^2 z}\right )}{a}\right )\right \}\right \} \]
Maple ✗
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,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,sigma,lambda1,lambda2,n1,n2,nu'); pde := a*x^2*(a*b*x*y-c^2*z^2)*diff(w(x,y,z),x)+a*x*y*(a*b*x*y-c^2*z^2)*diff(w(x,y,z),y) + b*y*z*(b*c*y*z+2*a^2*x^2)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ \text { sol=() } \]
____________________________________________________________________________________
Added April 15, 2019.
Problem Chapter 6.2.3.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x(c z^4 - b y^4) w_x + y(a x^4-2 c z^4) w_y + z(2 b y^4-a x^4) w_z= 0 \]
Mathematica ✗
ClearAll[w, x, y, z, 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,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = x*(c*z^4 - b*y^4)*D[w[x, y,z], x] +y*(a*x^4-2*c*z^4)*D[w[x, y,z], y] +z*(2*b*y^4-a*x^4) *D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \text {Failed} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,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,sigma,lambda1,lambda2,n1,n2,nu'); pde := x*(c*z^4 - b*y^4)*diff(w(x,y,z),x)+y*(a*x^4-2*c*z^4)*diff(w(x,y,z),y) + z*(2*b*y^4-a*x^4)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={{\rm e}^{-1/16\,{\it \_C1}\,{x}^{4}}}{{\rm e}^{{\it \_C2}/16}}{{\rm e}^{-1/16\,{\frac {{\it \_C1}\,b{y}^{4}}{a}}}}{x}^{{\it \_C2}/4}{\it \_C3}\,{\it \_C5}\,{y}^{{\it \_C2}/8}{\it \_C4}\,{z}^{{\it \_C2}/8}{{\rm e}^{-1/16\,{\frac {{\it \_C1}\,c{z}^{4}}{a}}}} \]
____________________________________________________________________________________
Added April 15, 2019.
Problem Chapter 6.2.3.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + y w_y + a \sqrt {x^2+y^2}w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, 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,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = x*D[w[x, y,z], x] +y*D[w[x, y,z], y] +a*Sqrt[x^2+y^2]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (\frac {y}{x},z-a \sqrt {x^2+y^2}\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,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,sigma,lambda1,lambda2,n1,n2,nu'); pde := x*diff(w(x,y,z),x)+y*diff(w(x,y,z),y) + a*sqrt(x^2+y^2)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {y}{x}},-a\sqrt {{x}^{2}+{y}^{2}}+z \right ) \]
____________________________________________________________________________________
Added April 15, 2019.
Problem Chapter 6.2.3.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + y w_y + (z- a \sqrt {x^2+y^2+z^2})w_z= 0 \]
Mathematica ✓
ClearAll[w, x, y, z, 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,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = x*D[w[x, y,z], x] +y*D[w[x, y,z], y] +(z-a*Sqrt[x^2+y^2+z^2])*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y,z)\to c_1\left (\frac {y}{x},\log \left (-\sqrt {\frac {x^{2 a} \left (y^2+2 z^2\right )+x^{2 a+2}-2 \sqrt {z^2 x^{4 a} \left (x^2+y^2+z^2\right )}}{x^2+y^2}}\right )\right )\right \},\left \{w(x,y,z)\to c_1\left (\frac {y}{x},\frac {1}{2} \log \left (\frac {x^{2 a} \left (y^2+2 z^2\right )+x^{2 a+2}-2 \sqrt {z^2 x^{4 a} \left (x^2+y^2+z^2\right )}}{x^2+y^2}\right )\right )\right \},\left \{w(x,y,z)\to c_1\left (\frac {y}{x},\log \left (-\sqrt {\frac {x^{2 a} \left (y^2+2 z^2\right )+x^{2 a+2}+2 \sqrt {z^2 x^{4 a} \left (x^2+y^2+z^2\right )}}{x^2+y^2}}\right )\right )\right \},\left \{w(x,y,z)\to c_1\left (\frac {y}{x},\frac {1}{2} \log \left (\frac {x^{2 a} \left (y^2+2 z^2\right )+x^{2 a+2}+2 \sqrt {z^2 x^{4 a} \left (x^2+y^2+z^2\right )}}{x^2+y^2}\right )\right )\right \}\right \} \]
Maple ✓
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,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,sigma,lambda1,lambda2,n1,n2,nu'); pde := x*diff(w(x,y,z),x)+y*diff(w(x,y,z),y) + (z-a*sqrt(x^2+y^2+z^2))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {y}{x}}, \left ( z+\sqrt {{x}^{2}+{y}^{2}+{z}^{2}} \right ) {x}^{a-1} \right ) \]
____________________________________________________________________________________
Added April 15, 2019.
Problem Chapter 6.2.3.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ z \sqrt {y^2+z^2} w_x + a z \sqrt {x^2+z^2} w_y - (x \sqrt {y^2+z^2}+a y \sqrt {x^2+z^2})w_z= 0 \]
Mathematica ✗
ClearAll[w, x, y, z, 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,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; pde = z*Sqrt[y^2+z^2]*D[w[x, y,z], x] +a*z*Sqrt[x^2+z^2]*D[w[x, y,z], y] -(x*Sqrt[y^2+z^2]+a*y*Sqrt[x^2+z^2])*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \text {Failed} \]
Maple ✗
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,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,sigma,lambda1,lambda2,n1,n2,nu'); pde := z*sqrt(y^2+z^2)*diff(w(x,y,z),x)+a*z*sqrt(x^2+z^2)*diff(w(x,y,z),y) -(x*sqrt(y^2+z^2)+a*y*sqrt(x^2+z^2))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ \text { sol=() } \]
____________________________________________________________________________________
Added April 15, 2019.
Problem Chapter 6.2.3.13, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (y-z) \sqrt {f(x)} w_x + (z-x)\sqrt {f(y)} w_y + (x-y)\sqrt {f(z)}w_z= 0 \] Where \[ f(t) = a_6 t^6 + a_5 t^5 + a_4 t^4 + a_3 t^3+ a_2 t^2 + a_1 t+a_0 \]
Mathematica ✗
ClearAll[w, x, y, z, 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,F,C1]; ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; f[t_]:= a[6]*t^6+a[5]*t^5+a[4]*t^4+a[3]*t^3+a[2]*t^2+a[1]*t+a[0]; pde = (y-z)*Sqrt[f[x]]*D[w[x, y,z], x] +(z-x)*Sqrt[f[y]]*D[w[x, y,z], y] +(x-y)*Sqrt[f[z]]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]]; sol = Simplify[sol];
\[ \text {Failed} \]
Maple ✗
unassign('w,x,y,z,a,b,n,m,c,k,alpha,beta,g,A,C1,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,sigma,lambda1,lambda2,n1,n2,nu'); f := t-> a[6]*t^6+a[5]*t^5+a[4]*t^4+a[3]*t^3+a[2]*t^2+a[1]*t+a[0]; pde := (y-z)*sqrt(f(x))*diff(w(x,y,z),x)+(z-x)*sqrt(f(y))*diff(w(x,y,z),y)+(x-y)*sqrt(f(z))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime')); if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
\[ \text { Exception } \]