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]];
\[\left \{\left \{w(x,y)\to \frac {e^{\frac {\alpha y}{b}} \left (-\beta \sqrt {x y} \left (\frac {\alpha y}{b}\right )^{-\frac {a+b}{2 b}} \text {Gamma}\left (\frac {a+b}{2 b},\frac {\alpha y}{b}\right )+b c_1\left (y x^{-\frac {b}{a}}\right )+\gamma \text {Ei}\left (-\frac {\alpha y}{b}\right )\right )}{b}\right \}\right \}\]
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'));
\[w \left ( x,y \right ) =-{\frac {1}{b \left ( 5\,b+a \right ) y \left ( 3\,b+a \right ) \alpha \, \left ( a+b \right ) a} \left ( -4\,a{x}^{-1/2\,{\frac {a+b}{a}}}{x}^{1/2\,{\frac {b}{a}}+1/2}{{\rm e}^{-1/2\,{\frac {\alpha \,y}{b}}}} \left ( {\frac {\alpha \,y}{b}} \right ) ^{-1/4\,{\frac {3\,b+a}{b}}}{b}^{3}\sqrt {xy} \left ( {\frac {\alpha \,y}{b}{x}^{-{\frac {b}{a}}}} \right ) ^{-1/2-1/2\,{\frac {a}{b}}} \left ( {\frac {\alpha \,y}{b}{x}^{-{\frac {b}{a}}}} \right ) ^{1/2+1/2\,{\frac {a}{b}}}\beta \, \left ( 2\,\alpha \,y+a+3\,b \right ) \WhittakerM \left ( 1/4\,{\frac {-b+a}{b}},1/4\,{\frac {5\,b+a}{b}},{\frac {\alpha \,y}{b}} \right ) + \left ( 3\,b+a \right ) \left ( -2\,a{x}^{-1/2\,{\frac {a+b}{a}}}{x}^{1/2\,{\frac {b}{a}}+1/2}{{\rm e}^{-1/2\,{\frac {\alpha \,y}{b}}}} \left ( {\frac {\alpha \,y}{b}} \right ) ^{-1/4\,{\frac {3\,b+a}{b}}}{b}^{2}\sqrt {xy} \left ( {\frac {\alpha \,y}{b}{x}^{-{\frac {b}{a}}}} \right ) ^{-1/2-1/2\,{\frac {a}{b}}} \left ( {\frac {\alpha \,y}{b}{x}^{-{\frac {b}{a}}}} \right ) ^{1/2+1/2\,{\frac {a}{b}}}\beta \, \left ( 3\,b+a \right ) \WhittakerM \left ( 1/4\,{\frac {3\,b+a}{b}},1/4\,{\frac {5\,b+a}{b}},{\frac {\alpha \,y}{b}} \right ) +\alpha \, \left ( a+b \right ) y \left ( 5\,b+a \right ) \left ( -a\gamma \,\ln \left ( {\frac {\alpha \,y}{b}{x}^{-{\frac {b}{a}}}} \right ) -ab{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) +\gamma \, \left ( a\ln \left ( {\frac {\alpha \,y}{b}} \right ) +a\Ei \left ( 1,{\frac {\alpha \,y}{b}} \right ) -b\ln \left ( x \right ) \right ) \right ) \right ) \right ) {{\rm e}^{{\frac {\alpha \,y}{b}}}}}\]
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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'));
\[w \left ( x,y \right ) =-{\frac {1}{2\,{\lambda }^{2} \left ( a+b \right ) } \left ( \beta \, \left ( a+b \right ) \left ( 2\,\sqrt {xy}\lambda +a+b \right ) {{\rm e}^{-2\,{\frac {\sqrt {xy}\lambda }{a+b}}}}-2\, \left ( -2\,\gamma \,\Ei \left ( 1,2\,{\frac {\sqrt {xy}\lambda }{a+b}} \right ) +{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) \left ( a+b \right ) \right ) {\lambda }^{2} \right ) {{\rm e}^{2\,{\frac {\sqrt {xy}\lambda }{a+b}}}}}\]
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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'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{ \left ( \beta \,\sqrt {{\it \_a}}+\gamma \right ) \left ( { \left ( {\it \_a}\,ab+\sqrt { \left ( a{y}^{2}+ \left ( {{\it \_a}}^{2}-{x}^{2} \right ) b \right ) a}\sqrt {ba} \right ) {\frac {1}{\sqrt {ba}}}} \right ) ^{-{\alpha {\frac {1}{\sqrt {ba}}}}}{\frac {1}{\sqrt { \left ( a{y}^{2}+ \left ( {{\it \_a}}^{2}-{x}^{2} \right ) b \right ) a}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {a{y}^{2}-b{x}^{2}}{a}} \right ) \right ) \left ( {abx{\frac {1}{\sqrt {ba}}}}+\sqrt {{a}^{2}{y}^{2}} \right ) ^{{\alpha {\frac {1}{\sqrt {ba}}}}}\]
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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'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{ \left ( \beta \,\sqrt {{\it \_a}}+\gamma \right ) \left ( { \left ( {\it \_a}\,ab+\sqrt { \left ( a{y}^{2}+ \left ( {{\it \_a}}^{2}-{x}^{2} \right ) b \right ) a}\sqrt {ba} \right ) {\frac {1}{\sqrt {ba}}}} \right ) ^{-{\alpha {\frac {1}{\sqrt {ba}}}}}{\frac {1}{\sqrt { \left ( a{y}^{2}+ \left ( {{\it \_a}}^{2}-{x}^{2} \right ) b \right ) a}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {a{y}^{2}-b{x}^{2}}{a}} \right ) \right ) \left ( {abx{\frac {1}{\sqrt {ba}}}}+\sqrt {{a}^{2}{y}^{2}} \right ) ^{{\alpha {\frac {1}{\sqrt {ba}}}}}\]
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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'));
\[w \left ( x,y \right ) =-{\frac {1}{2\,{\alpha }^{3}}{{\rm e}^{2\,{\frac {\sqrt {y}\alpha }{b}}}} \left ( -2\,{\it \_F1} \left ( {\frac {-\sqrt {y}a+\sqrt {x}b}{b}} \right ) {\alpha }^{3}+ \left ( 2\,a\beta \,\alpha \,\sqrt {x}+2\,\sqrt {y}b\alpha \,\gamma + \left ( 2\,\beta \,x+2\,\gamma \,y+2\,\delta \right ) {\alpha }^{2}+{a}^{2}\beta +\gamma \,{b}^{2} \right ) {{\rm e}^{-2\,{\frac {\sqrt {y}\alpha }{b}}}} \right ) }\]
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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'));
\[w \left ( x,y \right ) = \left ( \int ^{y}\!{\frac {1}{b}{{\rm e}^{-2\,{\frac {\sqrt {{\it \_a}}\alpha }{b}}}} \left ( {\it \_a}\,\gamma +\beta \,\sqrt {{\frac {1}{{b}^{2}} \left ( \sqrt {{\it \_a}}a-\sqrt {y}a+\sqrt {x}b \right ) ^{2}}}+\delta \right ) {\frac {1}{\sqrt {{\it \_a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {1}{b} \left ( -\sqrt {y}a+\sqrt {x}b \right ) } \right ) \right ) {{\rm e}^{2\,{\frac {\sqrt {y}\alpha }{b}}}}\]
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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'));
\[w \left ( x,y \right ) = \left ( \int ^{y}\!{\frac {1}{b}{{\rm e}^{-{\frac {\alpha }{b}\int \!{\frac {1}{\sqrt {{\frac {1}{{b}^{2}} \left ( \left ( {{\it \_b}}^{{\frac {3}{2}}}a+\RootOf \left ( x{b}^{2}- \left ( {b}^{2}{y}^{{\frac {3}{2}}}a+{b}^{3}{\it \_Z} \right ) ^{{\frac {2}{3}}} \right ) b \right ) {b}^{2} \right ) ^{{\frac {2}{3}}}}}}}\,{\rm d}{\it \_b}}}} \left ( \gamma \,{\it \_b}+\beta \,\sqrt {{\frac {1}{{b}^{2}} \left ( \left ( {{\it \_b}}^{{\frac {3}{2}}}a+\RootOf \left ( x{b}^{2}- \left ( {b}^{2}{y}^{{\frac {3}{2}}}a+{b}^{3}{\it \_Z} \right ) ^{{\frac {2}{3}}} \right ) b \right ) {b}^{2} \right ) ^{{\frac {2}{3}}}}}+\delta \right ) {\frac {1}{\sqrt {{\frac {1}{{b}^{2}} \left ( \left ( {{\it \_b}}^{{\frac {3}{2}}}a+\RootOf \left ( x{b}^{2}- \left ( {b}^{2}{y}^{{\frac {3}{2}}}a+{b}^{3}{\it \_Z} \right ) ^{{\frac {2}{3}}} \right ) b \right ) {b}^{2} \right ) ^{{\frac {2}{3}}}}}}}}{d{\it \_b}}+{\it \_F1} \left ( \RootOf \left ( x{b}^{2}- \left ( {b}^{2}{y}^{{\frac {3}{2}}}a+{b}^{3}{\it \_Z} \right ) ^{{\frac {2}{3}}} \right ) \right ) \right ) {{\rm e}^{\int ^{y}\!{\frac {\alpha }{b}{\frac {1}{\sqrt {{\frac {1}{{b}^{2}} \left ( {b}^{2}{{\it \_a}}^{{\frac {3}{2}}}a+{b}^{3}\RootOf \left ( x{b}^{2}- \left ( {b}^{2}{y}^{{\frac {3}{2}}}a+{b}^{3}{\it \_Z} \right ) ^{{\frac {2}{3}}} \right ) \right ) ^{{\frac {2}{3}}}}}}}}{d{\it \_a}}}}\] contains RootOf
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