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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to \frac {e^{\frac {\alpha y}{b}} \left (-\beta \sqrt {x y} \left (\frac {\alpha y}{b}\right )^{-\frac {a+b}{2 b}} \operatorname {Gamma}\left (\frac {a+b}{2 b},\frac {\alpha y}{b}\right )+b c_1\left (y x^{-\frac {b}{a}}\right )+\gamma \text {Ei}\left (-\frac {\alpha y}{b}\right )\right )}{b}\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = -\frac {\left (-4 \sqrt {x y}\, \left (2 y \alpha +a +3 b \right ) a b^{3} \beta x^{-\frac {a +b}{2 a}} x^{\frac {b}{2 a}+\frac {1}{2}} \left (\frac {\alpha y}{b}\right )^{-\frac {a +3 b}{4 b}} \left (\frac {\alpha y x^{-\frac {b}{a}}}{b}\right )^{-\frac {a}{2 b}-\frac {1}{2}} \left (\frac {\alpha y x^{-\frac {b}{a}}}{b}\right )^{\frac {a}{2 b}+\frac {1}{2}} \WhittakerM \left (\frac {a -b}{4 b}, \frac {a +5 b}{4 b}, \frac {\alpha y}{b}\right ) {\mathrm e}^{-\frac {\alpha y}{2 b}}+\left (-2 \sqrt {x y}\, \left (a +3 b \right ) a b^{2} \beta x^{-\frac {a +b}{2 a}} x^{\frac {b}{2 a}+\frac {1}{2}} \left (\frac {\alpha y}{b}\right )^{-\frac {a +3 b}{4 b}} \left (\frac {\alpha y x^{-\frac {b}{a}}}{b}\right )^{-\frac {a}{2 b}-\frac {1}{2}} \left (\frac {\alpha y x^{-\frac {b}{a}}}{b}\right )^{\frac {a}{2 b}+\frac {1}{2}} \WhittakerM \left (\frac {a +3 b}{4 b}, \frac {a +5 b}{4 b}, \frac {\alpha y}{b}\right ) {\mathrm e}^{-\frac {\alpha y}{2 b}}+\left (a +b \right ) \left (-a b \mathit {\_F1} \left (y x^{-\frac {b}{a}}\right )-\gamma a \ln \left (\frac {\alpha y x^{-\frac {b}{a}}}{b}\right )+\gamma \left (a \Ei \left (1, \frac {\alpha y}{b}\right )+a \ln \left (\frac {\alpha y}{b}\right )-b \ln \left (x \right )\right )\right ) \left (a +5 b \right ) \alpha y \right ) \left (a +3 b \right )\right ) {\mathrm e}^{\frac {\alpha y}{b}}}{\left (a +5 b \right ) \left (a +3 b \right ) \left (a +b \right ) a \alpha b y}\]
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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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to e^{\frac {2 \lambda \sqrt {x y}}{a+b}} \left (\int _1^x\frac {e^{-\frac {2 \lambda \sqrt {x^{-\frac {b}{a}} y K[1]^{\frac {a+b}{a}}}}{a+b}} \left (\beta y K[1]^{\frac {a+b}{a}} x^{-\frac {b}{a}}+\gamma \right )}{a K[1]}dK[1]+c_1\left (y x^{-\frac {b}{a}}\right )\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = -\frac {\left (\left (a +b \right ) \left (a +b +2 \sqrt {x y}\, \lambda \right ) \beta \,{\mathrm e}^{-\frac {2 \sqrt {x y}\, \lambda }{a +b}}-2 \left (-2 \gamma \Ei \left (1, \frac {2 \sqrt {x y}\, \lambda }{a +b}\right )+\left (a +b \right ) \mathit {\_F1} \left (y x^{-\frac {b}{a}}\right )\right ) \lambda ^{2}\right ) {\mathrm e}^{\frac {2 \sqrt {x y}\, \lambda }{a +b}}}{2 \left (a +b \right ) \lambda ^{2}}\]
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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["Global`*"]; 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]];
\begin {align*} & \left \{w(x,y)\to e^{-\frac {\alpha \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a y^2}}\right )}{\sqrt {a} \sqrt {b}}} \left (\int _1^x-\frac {\exp \left (\frac {\alpha \tanh ^{-1}\left (\frac {\sqrt {b} K[1]}{\sqrt {-b x^2+a y^2+b K[1]^2}}\right )}{\sqrt {a} \sqrt {b}}\right ) \left (\sqrt {K[1]} \beta +\gamma \right )}{\sqrt {a} \sqrt {a y^2+b \left (K[1]^2-x^2\right )}}dK[1]+c_1\left (\frac {a y^2-b x^2}{2 a}\right )\right )\right \}\\& \left \{w(x,y)\to e^{\frac {\alpha \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a y^2}}\right )}{\sqrt {a} \sqrt {b}}} \left (\int _1^x\frac {\exp \left (-\frac {\alpha \tanh ^{-1}\left (\frac {\sqrt {b} K[2]}{\sqrt {-b x^2+a y^2+b K[2]^2}}\right )}{\sqrt {a} \sqrt {b}}\right ) \left (\sqrt {K[2]} \beta +\gamma \right )}{\sqrt {a} \sqrt {a y^2+b \left (K[2]^2-x^2\right )}}dK[2]+c_1\left (\frac {a y^2-b x^2}{2 a}\right )\right )\right \}\\ \end {align*}
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\sqrt {\mathit {\_a}}\, \beta +\gamma \right ) \left (\frac {\mathit {\_a} a b +\sqrt {\left (a y^{2}+\left (\mathit {\_a}^{2}-x^{2}\right ) b \right ) a}\, \sqrt {a b}}{\sqrt {a b}}\right )^{-\frac {\alpha }{\sqrt {a b}}}}{\sqrt {\left (a y^{2}+\left (\mathit {\_a}^{2}-x^{2}\right ) b \right ) a}}d\mathit {\_a} +\mathit {\_F1} \left (\frac {a y^{2}-b x^{2}}{a}\right )\right ) \left (\frac {a b x}{\sqrt {a b}}+\sqrt {a^{2} y^{2}}\right )^{\frac {\alpha }{\sqrt {a b}}}\]
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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["Global`*"]; 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]];
\begin {align*} & \left \{w(x,y)\to e^{-\frac {\alpha \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a y^2}}\right )}{\sqrt {a} \sqrt {b}}} \left (\int _1^x-\frac {\exp \left (\frac {\alpha \tanh ^{-1}\left (\frac {\sqrt {b} K[1]}{\sqrt {-b x^2+a y^2+b K[1]^2}}\right )}{\sqrt {a} \sqrt {b}}\right ) \left (\sqrt {K[1]} \beta +\gamma \right )}{\sqrt {a} \sqrt {a y^2+b \left (K[1]^2-x^2\right )}}dK[1]+c_1\left (\frac {a y^2-b x^2}{2 a}\right )\right )\right \}\\& \left \{w(x,y)\to e^{\frac {\alpha \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a y^2}}\right )}{\sqrt {a} \sqrt {b}}} \left (\int _1^x\frac {\exp \left (-\frac {\alpha \tanh ^{-1}\left (\frac {\sqrt {b} K[2]}{\sqrt {-b x^2+a y^2+b K[2]^2}}\right )}{\sqrt {a} \sqrt {b}}\right ) \left (\sqrt {K[2]} \beta +\gamma \right )}{\sqrt {a} \sqrt {a y^2+b \left (K[2]^2-x^2\right )}}dK[2]+c_1\left (\frac {a y^2-b x^2}{2 a}\right )\right )\right \}\\ \end {align*}
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\sqrt {\mathit {\_a}}\, \beta +\gamma \right ) \left (\frac {\mathit {\_a} a b +\sqrt {\left (a y^{2}+\left (\mathit {\_a}^{2}-x^{2}\right ) b \right ) a}\, \sqrt {a b}}{\sqrt {a b}}\right )^{-\frac {\alpha }{\sqrt {a b}}}}{\sqrt {\left (a y^{2}+\left (\mathit {\_a}^{2}-x^{2}\right ) b \right ) a}}d\mathit {\_a} +\mathit {\_F1} \left (\frac {a y^{2}-b x^{2}}{a}\right )\right ) \left (\frac {a b x}{\sqrt {a b}}+\sqrt {a^{2} y^{2}}\right )^{\frac {\alpha }{\sqrt {a b}}}\]
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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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to -\frac {-2 \alpha ^3 e^{\frac {2 \alpha \sqrt {x}}{a}} c_1\left (2 \sqrt {y}-\frac {2 b \sqrt {x}}{a}\right )+a^2 \beta +2 a \alpha \beta \sqrt {x}+2 \alpha ^2 \beta x+2 \alpha ^2 \delta +2 \alpha ^2 \gamma y+2 \alpha b \gamma \sqrt {y}+b^2 \gamma }{2 \alpha ^3}\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = -\frac {\left (-2 \alpha ^{3} \mathit {\_F1} \left (\frac {-a \sqrt {y}+b \sqrt {x}}{b}\right )+\left (2 a \alpha \beta \sqrt {x}+a^{2} \beta +2 \gamma \alpha b \sqrt {y}+\left (2 \beta x +2 \delta +2 \gamma y \right ) \alpha ^{2}+\gamma b^{2}\right ) {\mathrm e}^{-\frac {2 \alpha \sqrt {y}}{b}}\right ) {\mathrm e}^{\frac {2 \alpha \sqrt {y}}{b}}}{2 \alpha ^{3}}\]
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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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to -\frac {a \beta +2 \alpha \left (\beta \sqrt {x}+\gamma \right )}{2 \alpha ^2}+e^{\frac {2 \alpha \sqrt {x}}{a}} c_1\left (2 \sqrt {y}-\frac {2 b \sqrt {x}}{a}\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \left (\int _{}^{y}\frac {\left (\gamma \mathit {\_a} +\sqrt {\frac {\left (\sqrt {\mathit {\_a}}\, a -a \sqrt {y}+b \sqrt {x}\right )^{2}}{b^{2}}}\, \beta +\delta \right ) {\mathrm e}^{-\frac {2 \sqrt {\mathit {\_a}}\, \alpha }{b}}}{\sqrt {\mathit {\_a}}\, b}d\mathit {\_a} +\mathit {\_F1} \left (\frac {-a \sqrt {y}+b \sqrt {x}}{b}\right )\right ) {\mathrm e}^{\frac {2 \alpha \sqrt {y}}{b}}\]
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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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to \exp \left (\frac {\alpha x \sqrt [3]{\frac {a y^{3/2}}{a y^{3/2}-b x^{3/2}}} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};\frac {b x^{3/2}}{b x^{3/2}-a y^{3/2}}\right )}{a \sqrt [3]{y^{3/2}}}\right ) \left (\int _1^x\frac {\exp \left (-\frac {\alpha \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};\frac {b K[1]^{3/2}}{b x^{3/2}-a y^{3/2}}\right ) K[1] \sqrt [3]{1-\frac {b K[1]^{3/2}}{b x^{3/2}-a y^{3/2}}}}{a \sqrt [3]{y^{3/2}+\frac {b \left (K[1]^{3/2}-x^{3/2}\right )}{a}}}\right ) \left (\sqrt {K[1]} \beta +\gamma \right )}{a \sqrt [3]{y^{3/2}+\frac {b \left (K[1]^{3/2}-x^{3/2}\right )}{a}}}dK[1]+c_1\left (\frac {2 \left (a y^{3/2}-b x^{3/2}\right )}{3 a}\right )\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \left (\int _{}^{y}\frac {\left (\gamma \mathit {\_b} +\sqrt {\frac {\left (\left (\mathit {\_b}^{\frac {3}{2}} a +b \RootOf \left (b^{2} x -\left (a b^{2} y^{\frac {3}{2}}+\mathit {\_Z} b^{3}\right )^{\frac {2}{3}}\right )\right ) b^{2}\right )^{\frac {2}{3}}}{b^{2}}}\, \beta +\delta \right ) {\mathrm e}^{-\frac {\alpha \left (\int \frac {1}{\sqrt {\frac {\left (\left (\mathit {\_b}^{\frac {3}{2}} a +b \RootOf \left (b^{2} x -\left (a b^{2} y^{\frac {3}{2}}+\mathit {\_Z} b^{3}\right )^{\frac {2}{3}}\right )\right ) b^{2}\right )^{\frac {2}{3}}}{b^{2}}}}d \mathit {\_b} \right )}{b}}}{\sqrt {\frac {\left (\left (\mathit {\_b}^{\frac {3}{2}} a +b \RootOf \left (b^{2} x -\left (a b^{2} y^{\frac {3}{2}}+\mathit {\_Z} b^{3}\right )^{\frac {2}{3}}\right )\right ) b^{2}\right )^{\frac {2}{3}}}{b^{2}}}\, b}d\mathit {\_b} +\mathit {\_F1} \left (\RootOf \left (b^{2} x -\left (a b^{2} y^{\frac {3}{2}}+\mathit {\_Z} b^{3}\right )^{\frac {2}{3}}\right )\right )\right ) {\mathrm e}^{\int _{}^{y}\frac {\alpha }{\sqrt {\frac {\left (\mathit {\_a}^{\frac {3}{2}} a b^{2}+b^{3} \RootOf \left (b^{2} x -\left (a b^{2} y^{\frac {3}{2}}+\mathit {\_Z} b^{3}\right )^{\frac {2}{3}}\right )\right )^{\frac {2}{3}}}{b^{2}}}\, b}d\mathit {\_a}}\] contains RootOf
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