2.3

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]];

\[\left \{\left \{w(x,y)\to \frac {e^{\frac {\alpha y}{b}} \left (b c_1\left (y x^{-\frac {b}{a}}\right )-\beta \sqrt {x y} \left (\frac {\alpha y}{b}\right )^{-\frac {a+b}{2 b}} \Gamma \left (\frac {a+b}{2 b},\frac {\alpha y}{b}\right )+\gamma \operatorname {ExpIntegralEi}\left (-\frac {\alpha y}{b}\right )\right )}{b}\right \}\right \}\]

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'));

\[w \left (x , y\right ) = \frac {4 a \sqrt {x y}\, {\mathrm e}^{\frac {\alpha y}{2 b}} \left (\frac {\alpha y}{b}\right )^{-\frac {3 b +a}{4 b}} b^{3} \beta \left (2 \alpha y +a +3 b \right ) \operatorname {WhittakerM}\left (\frac {a -b}{4 b}, \frac {5 b +a}{4 b}, \frac {\alpha y}{b}\right )-\left (3 b +a \right ) \left (-2 a \sqrt {x y}\, {\mathrm e}^{\frac {\alpha y}{2 b}} \left (\frac {\alpha y}{b}\right )^{-\frac {3 b +a}{4 b}} b^{2} \beta \left (3 b +a \right ) \operatorname {WhittakerM}\left (\frac {3 b +a}{4 b}, \frac {5 b +a}{4 b}, \frac {\alpha y}{b}\right )+{\mathrm e}^{\frac {\alpha y}{b}} \left (-a \gamma \ln \left (\frac {\alpha y \,x^{-\frac {b}{a}}}{b}\right )-a b f_{1} \left (y \,x^{-\frac {b}{a}}\right )+\gamma \left (a \ln \left (\frac {\alpha y}{b}\right )+a \,\operatorname {Ei}_{1}\left (\frac {\alpha y}{b}\right )-b \ln \left (x \right )\right )\right ) y \left (5 b +a \right ) \left (a +b \right ) \alpha \right )}{b \alpha y \left (5 b +a \right ) \left (a +b \right ) \left (3 b +a \right ) a}\]

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'));

\[w \left (x , y\right ) = \frac {-4 \gamma \,\operatorname {Ei}_{1}\left (\frac {2 \sqrt {x y}\, \lambda }{a +b}\right ) \lambda ^{2} {\mathrm e}^{\frac {2 \sqrt {x y}\, \lambda }{a +b}}-\left (a +b \right ) \left (-2 f_{1} \left (y \,x^{-\frac {b}{a}}\right ) \lambda ^{2} {\mathrm e}^{\frac {2 \sqrt {x y}\, \lambda }{a +b}}+\beta \left (2 \sqrt {x y}\, \lambda +a +b \right )\right )}{2 \lambda ^{2} \left (a +b \right )}\]

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 \exp \left (-\frac {\alpha \sqrt {\frac {a y^2}{a y^2-b x^2}} \text {arcsinh}\left (x \sqrt {\frac {b}{a y^2-b x^2}}\right )}{\sqrt {a} \sqrt {a y^2} \sqrt {\frac {b}{a y^2-b x^2}}}\right ) \left (\int _1^x-\frac {\exp \left (\frac {\alpha \text {arcsinh}\left (\sqrt {\frac {b}{a y^2-b x^2}} K[1]\right ) \sqrt {\frac {2 b K[1]^2}{a y^2-b x^2}+2}}{\sqrt {2} \sqrt {a} \sqrt {\frac {b}{a y^2-b x^2}} \sqrt {a y^2+b \left (K[1]^2-x^2\right )}}\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 \exp \left (\frac {\alpha \sqrt {\frac {a y^2}{a y^2-b x^2}} \text {arcsinh}\left (x \sqrt {\frac {b}{a y^2-b x^2}}\right )}{\sqrt {a} \sqrt {a y^2} \sqrt {\frac {b}{a y^2-b x^2}}}\right ) \left (\int _1^x\frac {\exp \left (-\frac {\alpha \text {arcsinh}\left (\sqrt {\frac {b}{a y^2-b x^2}} K[2]\right ) \sqrt {\frac {2 b K[2]^2}{a y^2-b x^2}+2}}{\sqrt {2} \sqrt {a} \sqrt {\frac {b}{a y^2-b x^2}} \sqrt {a y^2+b \left (K[2]^2-x^2\right )}}\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'));

\[w \left (x , y\right ) = {\left (\frac {a b x +\sqrt {y^{2} a^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right )}^{\frac {\alpha }{\sqrt {a b}}} \left (\int _{}^{x}\frac {\left (\beta \sqrt {\textit {\_a}}+\gamma \right ) {\left (\frac {a b \textit {\_a} +\sqrt {a b}\, \sqrt {a \left (\left (\textit {\_a}^{2}-x^{2}\right ) b +y^{2} a \right )}}{\sqrt {a b}}\right )}^{-\frac {\alpha }{\sqrt {a b}}}}{\sqrt {a \left (\left (\textit {\_a}^{2}-x^{2}\right ) b +y^{2} a \right )}}d \textit {\_a} +f_{1} \left (y^{2}-\frac {b \,x^{2}}{a}\right )\right )\]

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 \exp \left (-\frac {\alpha \sqrt {\frac {a y^2}{a y^2-b x^2}} \text {arcsinh}\left (x \sqrt {\frac {b}{a y^2-b x^2}}\right )}{\sqrt {a} \sqrt {a y^2} \sqrt {\frac {b}{a y^2-b x^2}}}\right ) \left (\int _1^x-\frac {\exp \left (\frac {\alpha \text {arcsinh}\left (\sqrt {\frac {b}{a y^2-b x^2}} K[1]\right ) \sqrt {\frac {2 b K[1]^2}{a y^2-b x^2}+2}}{\sqrt {2} \sqrt {a} \sqrt {\frac {b}{a y^2-b x^2}} \sqrt {a y^2+b \left (K[1]^2-x^2\right )}}\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 \exp \left (\frac {\alpha \sqrt {\frac {a y^2}{a y^2-b x^2}} \text {arcsinh}\left (x \sqrt {\frac {b}{a y^2-b x^2}}\right )}{\sqrt {a} \sqrt {a y^2} \sqrt {\frac {b}{a y^2-b x^2}}}\right ) \left (\int _1^x\frac {\exp \left (-\frac {\alpha \text {arcsinh}\left (\sqrt {\frac {b}{a y^2-b x^2}} K[2]\right ) \sqrt {\frac {2 b K[2]^2}{a y^2-b x^2}+2}}{\sqrt {2} \sqrt {a} \sqrt {\frac {b}{a y^2-b x^2}} \sqrt {a y^2+b \left (K[2]^2-x^2\right )}}\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'));

\[w \left (x , y\right ) = {\left (\frac {a b x +\sqrt {y^{2} a^{2}}\, \sqrt {a b}}{\sqrt {a b}}\right )}^{\frac {\alpha }{\sqrt {a b}}} \left (\int _{}^{x}\frac {\left (\beta \sqrt {\textit {\_a}}+\gamma \right ) {\left (\frac {a b \textit {\_a} +\sqrt {a b}\, \sqrt {a \left (\left (\textit {\_a}^{2}-x^{2}\right ) b +y^{2} a \right )}}{\sqrt {a b}}\right )}^{-\frac {\alpha }{\sqrt {a b}}}}{\sqrt {a \left (\left (\textit {\_a}^{2}-x^{2}\right ) b +y^{2} a \right )}}d \textit {\_a} +f_{1} \left (y^{2}-\frac {b \,x^{2}}{a}\right )\right )\]

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'));

\[w \left (x , y\right ) = \frac {2 f_{1} \left (\frac {-\sqrt {y}\, a +b \sqrt {x}}{b}\right ) \alpha ^{3} {\mathrm e}^{\frac {2 \sqrt {y}\, \alpha }{b}}-2 \sqrt {x}\, a \alpha \beta -2 \sqrt {y}\, \alpha b \gamma +\left (-2 \beta x -2 \gamma y -2 \delta \right ) \alpha ^{2}-a^{2} \beta -b^{2} \gamma }{2 \alpha ^{3}}\]

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'));

\[w \left (x , y\right ) = \frac {\left (f_{1} \left (\frac {-\sqrt {y}\, a +b \sqrt {x}}{b}\right ) b +\int _{}^{y}\frac {{\mathrm e}^{-\frac {2 \sqrt {\textit {\_a}}\, \alpha }{b}} \left (\beta \sqrt {\frac {\left (\sqrt {\textit {\_a}}\, a -\sqrt {y}\, a +b \sqrt {x}\right )^{2}}{b^{2}}}+\gamma \textit {\_a} +\delta \right )}{\sqrt {\textit {\_a}}}d \textit {\_a} \right ) {\mathrm e}^{\frac {2 \sqrt {y}\, \alpha }{b}}}{b}\]

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]];

\[\left \{\left \{w(x,y)\to \exp \left (\frac {\alpha x \sqrt [3]{\frac {a y^{3/2}}{a y^{3/2}-b x^{3/2}}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {b x^{3/2}}{b x^{3/2}-a y^{3/2}}\right )}{a \sqrt [3]{y^{3/2}}}\right ) \left (\int _1^x\frac {\exp \left (-\frac {\alpha \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {b K[1]^{3/2}}{b x^{3/2}-a y^{3/2}}\right ) K[1] \sqrt [3]{1-\frac {b K[1]^{3/2}}{b x^{3/2}-a y^{3/2}}}}{a \sqrt [3]{y^{3/2}+\frac {b \left (K[1]^{3/2}-x^{3/2}\right )}{a}}}\right ) \left (\sqrt {K[1]} \beta +\gamma \right )}{a \sqrt [3]{y^{3/2}+\frac {b \left (K[1]^{3/2}-x^{3/2}\right )}{a}}}dK[1]+c_1\left (\frac {2 \left (a y^{3/2}-b x^{3/2}\right )}{3 a}\right )\right )\right \}\right \}\]

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'));

\[w \left (x , y\right ) = \frac {\left (f_{1} \left (\operatorname {RootOf}\left (x \,b^{2}-\left (b^{2} \left (y^{{3}/{2}} a +\textit {\_Z} b \right )\right )^{{2}/{3}}\right )\right ) b +\int _{}^{y}\frac {{\mathrm e}^{-\frac {\alpha \int \frac {1}{\sqrt {\frac {\left (b^{2} \left (\textit {\_a}^{{3}/{2}} a +\operatorname {RootOf}\left (x \,b^{2}-\left (b^{2} \left (y^{{3}/{2}} a +\textit {\_Z} b \right )\right )^{{2}/{3}}\right ) b \right )\right )^{{2}/{3}}}{b^{2}}}}d \textit {\_a}}{b}} \left (\beta \sqrt {\frac {\left (b^{2} \left (\textit {\_a}^{{3}/{2}} a +\operatorname {RootOf}\left (x \,b^{2}-\left (b^{2} \left (y^{{3}/{2}} a +\textit {\_Z} b \right )\right )^{{2}/{3}}\right ) b \right )\right )^{{2}/{3}}}{b^{2}}}+\gamma \textit {\_a} +\delta \right )}{\sqrt {\frac {\left (b^{2} \left (\textit {\_a}^{{3}/{2}} a +\operatorname {RootOf}\left (x \,b^{2}-\left (b^{2} \left (y^{{3}/{2}} a +\textit {\_Z} b \right )\right )^{{2}/{3}}\right ) b \right )\right )^{{2}/{3}}}{b^{2}}}}d \textit {\_a} \right ) {\mathrm e}^{\frac {\alpha \int _{}^{y}\frac {1}{\sqrt {\frac {\left (b^{2} \left (\textit {\_a}^{{3}/{2}} a +\operatorname {RootOf}\left (x \,b^{2}-\left (b^{2} \left (y^{{3}/{2}} a +\textit {\_Z} b \right )\right )^{{2}/{3}}\right ) b \right )\right )^{{2}/{3}}}{b^{2}}}}d \textit {\_a}}{b}}}{b}\]

contains RootOf

6.5.3 2.3

6.5.3.1 [1212] Problem 1

6.5.3.2 [1213] Problem 2

6.5.3.3 [1214] Problem 3

6.5.3.4 [1215] Problem 4

6.5.3.5 [1216] Problem 5

6.5.3.6 [1217] Problem 6

6.5.3.7 [1218] Problem 7