____________________________________________________________________________________
Added March 12, 2019.
Problem Chapter 5.2.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x w_x + b y w_y = \alpha y w + \beta \sqrt {x y} + \gamma \]
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, k22, k11, k12, s11, s22, s12]; pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == alpha*y*w[x, y] + beta*Sqrt[x*y] + gamma; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y)\to \frac {e^{\frac {\alpha y}{b}} \left (-\beta \sqrt {x y} \left (\frac {\alpha y}{b}\right )^{-\frac {a+b}{2 b}} \text {Gamma}\left (\frac {a+b}{2 b},\frac {\alpha y}{b}\right )+b c_1\left (y x^{-\frac {b}{a}}\right )+\gamma \text {ExpIntegralEi}\left (-\frac {\alpha y}{b}\right )\right )}{b}\right \}\right \} \]
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) = alpha*y*w(x,y)+ beta*sqrt(x*y)+gamma; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); sol:=simplify(sol);
\[ w \left ( x,y \right ) =-{\frac {1}{b \left ( 3\,b+a \right ) \left ( 5\,b+a \right ) \left ( a+b \right ) y\alpha \,a}{{\rm e}^{1/2\,{\frac {\alpha \,y}{b}}}} \left ( -4\,a\sqrt {yx} \left ( {\frac {\alpha \,y}{b}} \right ) ^{-1/4\,{\frac {3\,b+a}{b}}}{b}^{3}\beta \, \left ( 2\,\alpha \,y+a+3\,b \right ) \WhittakerM \left ( 1/4\,{\frac {a-b}{b}},1/4\,{\frac {5\,b+a}{b}},{\frac {\alpha \,y}{b}} \right ) + \left ( -2\,a\sqrt {yx} \left ( {\frac {\alpha \,y}{b}} \right ) ^{-1/4\,{\frac {3\,b+a}{b}}}{b}^{2}\beta \, \left ( 3\,b+a \right ) \WhittakerM \left ( 1/4\,{\frac {3\,b+a}{b}},1/4\,{\frac {5\,b+a}{b}},{\frac {\alpha \,y}{b}} \right ) +{{\rm e}^{1/2\,{\frac {\alpha \,y}{b}}}}y \left ( -\gamma \,\ln \left ( {\frac {\alpha \,y}{b}{x}^{-{\frac {b}{a}}}} \right ) a-ab{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) +\gamma \, \left ( \ln \left ( {\frac {\alpha \,y}{b}} \right ) a+a\Ei \left ( 1,{\frac {\alpha \,y}{b}} \right ) -b\ln \left ( x \right ) \right ) \right ) \alpha \, \left ( a+b \right ) \left ( 5\,b+a \right ) \right ) \left ( 3\,b+a \right ) \right ) } \]
____________________________________________________________________________________
Added March 12, 2019.
Problem Chapter 5.2.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x w_x + b y w_y = \lambda \sqrt {x y} w + \beta x y + \gamma \]
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, k22, k11, k12, s11, s22, s12]; pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == lambda*Sqrt[x*y]*w[x, y] + beta*x*y + gamma; 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*x*diff(w(x,y),x)+ b*y*diff(w(x,y),y) = lambda*sqrt(x*y)*w(x,y)+ beta*x*y+gamma; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); sol:=simplify(sol);
\[ w \left ( x,y \right ) =1/2\,{\frac {1}{{\lambda }^{2} \left ( a+b \right ) } \left ( -4\,\gamma \,\Ei \left ( 1,2\,{\frac {\sqrt {yx}\lambda }{a+b}} \right ) {\lambda }^{2}{{\rm e}^{2\,{\frac {\sqrt {yx}\lambda }{a+b}}}}- \left ( -2\,{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) {\lambda }^{2}{{\rm e}^{2\,{\frac {\sqrt {yx}\lambda }{a+b}}}}+\beta \, \left ( 2\,\sqrt {yx}\lambda +a+b \right ) \right ) \left ( a+b \right ) \right ) } \]
____________________________________________________________________________________
Added March 12, 2019.
Problem Chapter 5.2.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a y w_x + b x w_y = \alpha w + \beta \sqrt {x} + \gamma \]
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, k22, k11, k12, s11, s22, s12]; pde = a*y*D[w[x, y], x] + b*x*D[w[x, y], y] == alpha*w[x, y] + beta*Sqrt[x] + gamma; 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)+ b*x*diff(w(x,y),y) = alpha*w(x,y)+ beta*sqrt(x)+gamma; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {\beta \,\sqrt {{\it \_a}}+\gamma }{\sqrt {a \left ( {{\it \_a}}^{2}b+{y}^{2}a-b{x}^{2} \right ) }} \left ( {\frac {{\it \_a}\,ab+\sqrt {a \left ( {{\it \_a}}^{2}b+{y}^{2}a-b{x}^{2} \right ) }\sqrt {ab}}{\sqrt {ab}}} \right ) ^{-{\frac {\alpha }{\sqrt {ab}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {{y}^{2}a-b{x}^{2}}{a}} \right ) \right ) \left ( {\frac {axb}{\sqrt {ab}}}+\sqrt {ab{x}^{2}+ \left ( {y}^{2}a-b{x}^{2} \right ) a} \right ) ^{{\frac {\alpha }{\sqrt {ab}}}} \]
____________________________________________________________________________________
Added March 12, 2019.
Problem Chapter 5.2.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a y w_x + b x w_y = \alpha w + \beta \sqrt {x} + \gamma \]
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, k22, k11, k12, s11, s22, s12]; pde = a*y*D[w[x, y], x] + b*x*D[w[x, y], y] == alpha*w[x, y] + beta*Sqrt[x] + gamma; 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)+ b*x*diff(w(x,y),y) = alpha*w(x,y)+ beta*sqrt(x)+gamma; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); sol:=simplify(sol,size);
\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {\beta \,\sqrt {{\it \_a}}+\gamma }{\sqrt {a \left ( {y}^{2}a+ \left ( {{\it \_a}}^{2}-{x}^{2} \right ) b \right ) }} \left ( {\frac {{\it \_a}\,ab+\sqrt {a \left ( {y}^{2}a+ \left ( {{\it \_a}}^{2}-{x}^{2} \right ) b \right ) }\sqrt {ab}}{\sqrt {ab}}} \right ) ^{-{\frac {\alpha }{\sqrt {ab}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {{y}^{2}a-b{x}^{2}}{a}} \right ) \right ) \left ( {\frac {axb}{\sqrt {ab}}}+\sqrt {{a}^{2}{y}^{2}} \right ) ^{{\frac {\alpha }{\sqrt {ab}}}} \]
____________________________________________________________________________________
Added March 12, 2019.
Problem Chapter 5.2.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a \sqrt {x} w_x + b \sqrt {y} w_y = \alpha w + \beta x + \gamma y + \delta \]
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, k22, k11, k12, s11, s22, s12]; pde = a*Sqrt[x]*D[w[x, y], x] + b*Sqrt[y]*D[w[x, y], y] == alpha*w[x, y] + beta*x + gamma*y + delta; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {-a^2 \beta +2 \alpha ^3 e^{\frac {2 \alpha \sqrt {x}}{a}} c_1\left (\frac {2 \left (a \sqrt {y}-b \sqrt {x}\right )}{a}\right )-2 a \alpha \beta \sqrt {x}-2 \alpha ^2 \beta x-2 \alpha ^2 \delta -2 \alpha ^2 \gamma y-2 \alpha b \gamma \sqrt {y}-b^2 \gamma }{2 \alpha ^3}\right \}\right \} \]
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*sqrt(x)*diff(w(x,y),x)+ b*sqrt(y)*diff(w(x,y),y) = alpha*w(x,y)+ beta*x+gamma*y+delta; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); sol:=simplify(sol,size);
\[ w \left ( x,y \right ) =-1/2\,{\frac {1}{{\alpha }^{3}} \left ( -2\,{\it \_F1} \left ( {\frac {-\sqrt {y}a+b\sqrt {x}}{b}} \right ) {\alpha }^{3}+ \left ( 2\,a\beta \,\alpha \,\sqrt {x}+2\,\sqrt {y}b\alpha \,\gamma + \left ( 2\,\beta \,x+2\,\gamma \,y+2\,\delta \right ) {\alpha }^{2}+{a}^{2}\beta +\gamma \,{b}^{2} \right ) {{\rm e}^{-2\,{\frac {\sqrt {y}\alpha }{b}}}} \right ) {{\rm e}^{2\,{\frac {\sqrt {y}\alpha }{b}}}}} \]
____________________________________________________________________________________
Added March 12, 2019.
Problem Chapter 5.2.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a \sqrt {x} w_x + b \sqrt {y} w_y = \alpha w + \beta \sqrt {x} + \gamma \]
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, k22, k11, k12, s11, s22, s12]; pde = a*Sqrt[x]*D[w[x, y], x] + b*Sqrt[y]*D[w[x, y], y] == alpha*w[x, y] + beta*Sqrt[x] + gamma; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y)\to e^{\frac {2 \alpha \sqrt {x}}{a}} c_1\left (2 \sqrt {y}-\frac {2 b \sqrt {x}}{a}\right )-\frac {a \beta +2 \alpha \left (\beta \sqrt {x}+\gamma \right )}{2 \alpha ^2}\right \}\right \} \]
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*sqrt(x)*diff(w(x,y),x)+ b*sqrt(y)*diff(w(x,y),y) = alpha*w(x,y)+ beta*sqrt(x)+gamma*y+delta; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) = \left ( \int ^{y}\!{\frac {1}{b\sqrt {{\it \_a}}}{{\rm e}^{-2\,{\frac {\sqrt {{\it \_a}}\alpha }{b}}}} \left ( \beta \,\sqrt {{\frac { \left ( \sqrt {{\it \_a}}a-\sqrt {y}a+b\sqrt {x} \right ) ^{2}}{{b}^{2}}}}+{\it \_a}\,\gamma +\delta \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {-\sqrt {y}a+b\sqrt {x}}{b}} \right ) \right ) {{\rm e}^{2\,{\frac {\sqrt {y}\alpha }{b}}}} \]
____________________________________________________________________________________
Added March 12, 2019.
Problem Chapter 5.2.3.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a \sqrt {y} w_x + b \sqrt {x} w_y = \alpha w + \beta \sqrt {x} + \gamma \]
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, k22, k11, k12, s11, s22, s12]; pde = a*Sqrt[y]*D[w[x, y], x] + b*Sqrt[x]*D[w[x, y], y] == alpha*w[x, y] + beta*Sqrt[x] + gamma; 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*sqrt(y)*diff(w(x,y),x)+ b*sqrt(x)*diff(w(x,y),y) = alpha*w(x,y)+ beta*sqrt(x)+gamma*y+delta; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) = \left ( \int ^{y}\!{\frac {1}{b}{{\rm e}^{-{\frac {\alpha }{b}\int \!{\frac {1}{\sqrt {{\frac { \left ( \left ( {{\it \_b}}^{3/2}a+\RootOf \left ( x{b}^{2}- \left ( {b}^{2}{y}^{3/2}a+{\it \_Z}\,{b}^{3} \right ) ^{2/3} \right ) b \right ) {b}^{2} \right ) ^{2/3}}{{b}^{2}}}}}}\,{\rm d}{\it \_b}}}} \left ( \gamma \,{\it \_b}+\beta \,\sqrt {{\frac { \left ( \left ( {{\it \_b}}^{3/2}a+\RootOf \left ( x{b}^{2}- \left ( {b}^{2}{y}^{3/2}a+{\it \_Z}\,{b}^{3} \right ) ^{2/3} \right ) b \right ) {b}^{2} \right ) ^{2/3}}{{b}^{2}}}}+\delta \right ) {\frac {1}{\sqrt {{\frac { \left ( \left ( {{\it \_b}}^{3/2}a+\RootOf \left ( x{b}^{2}- \left ( {b}^{2}{y}^{3/2}a+{\it \_Z}\,{b}^{3} \right ) ^{2/3} \right ) b \right ) {b}^{2} \right ) ^{2/3}}{{b}^{2}}}}}}}{d{\it \_b}}+{\it \_F1} \left ( \RootOf \left ( x{b}^{2}- \left ( {b}^{2}{y}^{3/2}a+{\it \_Z}\,{b}^{3} \right ) ^{2/3} \right ) \right ) \right ) {{\rm e}^{\int ^{y}\!{\frac {\alpha }{b}{\frac {1}{\sqrt {{\frac { \left ( {b}^{2}{{\it \_a}}^{3/2}a+{b}^{3}\RootOf \left ( x{b}^{2}- \left ( {b}^{2}{y}^{3/2}a+{\it \_Z}\,{b}^{3} \right ) ^{2/3} \right ) \right ) ^{2/3}}{{b}^{2}}}}}}}{d{\it \_a}}}} \] contains RootOf