Added March 12, 2019.
Problem Chapter 5.2.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c w + k x^n y^m \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x, y] + k*x^n*y^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} k K[1]^n \left (y+\frac {b (K[1]-x)}{a}\right )^m}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+ k*x^n*y^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {k \,\textit {\_a}^{n} \left (\frac {a y -\left (-\textit {\_a} +x \right ) b}{a}\right )^{m} {\mathrm e}^{-\frac {\textit {\_a} c}{a}}}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]
____________________________________________________________________________________
Added March 12, 2019.
Problem Chapter 5.2.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + y w_y = b w + c x^n y^m \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + y*D[w[x, y], y] == b*w[x, y] + c*x^n*y^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {x (b-m)}{a}} \left (-\frac {c y^m x^n \left (\frac {x (b-m)}{a}\right )^{-n} \operatorname {Gamma}\left (n+1,\frac {x (b-m)}{a}\right )}{b-m}+e^{\frac {m x}{a}} c_1\left (y e^{-\frac {x}{a}}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ y*diff(w(x,y),y) = b*w(x,y)+ c*x^n*y^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {c \,\textit {\_a}^{n} \left (y \,{\mathrm e}^{\frac {\textit {\_a} -x}{a}}\right )^{m} {\mathrm e}^{-\frac {\textit {\_a} b}{a}}}{a}d \textit {\_a} +\textit {\_F1} \left (y \,{\mathrm e}^{-\frac {x}{a}}\right )\right ) {\mathrm e}^{\frac {b x}{a}}\]
____________________________________________________________________________________
Added April 1, 2019.
Problem Chapter 5.2.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + y w_y = a x w + b x^n y^m \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + y*D[w[x, y], y] == a*x*w[x, y] + b*x^n*y^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{a x} \left (-b y^m x^n (a x)^{-m-n} \operatorname {Gamma}(m+n,a x)+c_1\left (\frac {y}{x}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x)+ y*diff(w(x,y),y) = a*x*w(x,y)+ b*x^n*y^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \frac {b \,x^{n} y^{m} \left (a x \right )^{-\frac {m}{2}-\frac {n}{2}} \WhittakerM \left (\frac {m}{2}+\frac {n}{2}, \frac {m}{2}+\frac {n}{2}+\frac {1}{2}, a x \right ) {\mathrm e}^{\frac {a x}{2}}}{\left (m +n +1\right ) \left (m +n \right )}+\frac {b \,x^{n -1} y^{m} \left (a x \right )^{-\frac {m}{2}-\frac {n}{2}} \WhittakerM \left (\frac {m}{2}+\frac {n}{2}+1, \frac {m}{2}+\frac {n}{2}+\frac {1}{2}, a x \right ) {\mathrm e}^{\frac {a x}{2}}}{\left (m +n \right ) a}+\textit {\_F1} \left (\frac {y}{x}\right ) {\mathrm e}^{a x}\]
____________________________________________________________________________________
Added April 1, 2019.
Problem Chapter 5.2.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + y w_y = a \sqrt {x^2+y^2} w + b x^n y^m \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + y*D[w[x, y], y] == a*Sqrt[x^2+y^2]*w[x,y] + b*x^n*y^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{a \sqrt {x^2+y^2}} \left (\int _1^xb e^{-a \sqrt {\left (\frac {y^2}{x^2}+1\right ) K[1]^2}} K[1]^{n-1} \left (\frac {y K[1]}{x}\right )^mdK[1]+c_1\left (\frac {y}{x}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x)+ y*diff(w(x,y),y) = a*sqrt(x^2+y^2)*w(x,y)+ b*x^n*y^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \frac {\left (\left (m +n +1\right ) b \,x^{-m} x^{m +n} y^{m} \left (\sqrt {x^{2}+y^{2}}\, a \right )^{-\frac {m}{2}-\frac {n}{2}} \left (\frac {\sqrt {x^{2}+y^{2}}\, a}{x}\right )^{-m -n} \left (\frac {\sqrt {x^{2}+y^{2}}\, a}{x}\right )^{m +n} \WhittakerM \left (\frac {m}{2}+\frac {n}{2}+1, \frac {m}{2}+\frac {n}{2}+\frac {1}{2}, \sqrt {x^{2}+y^{2}}\, a \right ) {\mathrm e}^{-\frac {\sqrt {x^{2}+y^{2}}\, a}{2}}+\sqrt {x^{2}+y^{2}}\, \left (b \,x^{-m} x^{m +n} y^{m} \left (\sqrt {x^{2}+y^{2}}\, a \right )^{-\frac {m}{2}-\frac {n}{2}} \left (\frac {\sqrt {x^{2}+y^{2}}\, a}{x}\right )^{-m -n} \left (\frac {\sqrt {x^{2}+y^{2}}\, a}{x}\right )^{m +n} \WhittakerM \left (\frac {m}{2}+\frac {n}{2}, \frac {m}{2}+\frac {n}{2}+\frac {1}{2}, \sqrt {x^{2}+y^{2}}\, a \right ) {\mathrm e}^{-\frac {\sqrt {x^{2}+y^{2}}\, a}{2}}+\left (m +n +1\right ) \left (m +n \right ) \textit {\_F1} \left (\frac {y}{x}\right )\right ) a \right ) {\mathrm e}^{\sqrt {x^{2}+y^{2}}\, a}}{\sqrt {x^{2}+y^{2}}\, \left (m +n \right ) \left (m +n +1\right ) a}\]
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Added April 1, 2019.
Problem Chapter 5.2.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x w_x + b y w_y = c x^n y^m w + p x^k y^s \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == c*x^n*y^m*w[x,y] + p*x^k*y^s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c y^m x^n}{a n+b m}} \left (\int _1^x\frac {\exp \left (-\frac {c K[1]^n \left (x^{-\frac {b}{a}} y K[1]^{\frac {b}{a}}\right )^m}{b m+a n}\right ) p K[1]^{k-1} \left (x^{-\frac {b}{a}} y K[1]^{\frac {b}{a}}\right )^s}{a}dK[1]+c_1\left (y x^{-\frac {b}{a}}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*x*diff(w(x,y),x)+ b*y*diff(w(x,y),y) = c*x^n*y^m*w(x,y)+ p*x^k*y^s; 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 n +b m \right )^{2} \left (c \,x^{n} y^{m}+\left (k +n \right ) a +\left (m +s \right ) b \right ) p \,x^{\frac {b s +\left (k -n \right ) a}{a}} x^{-\frac {b s}{a}} y^{s} y^{-m} \left (\frac {c \,x^{n} y^{m}}{a n +b m}\right )^{\frac {\left (-k -n \right ) a -\left (m +s \right ) b}{2 a n +2 b m}} \left (\frac {c \,x^{-\frac {b m}{a}} y^{m}}{a n +b m}\right )^{\frac {-k a -b s}{a n +b m}} \left (\frac {c \,x^{-\frac {b m}{a}} y^{m}}{a n +b m}\right )^{\frac {k a +b s}{a n +b m}} \WhittakerM \left (\frac {\left (k -n \right ) a -\left (m -s \right ) b}{2 a n +2 b m}, \frac {\left (k +2 n \right ) a +2 \left (m +\frac {s}{2}\right ) b}{2 a n +2 b m}, \frac {c \,x^{n} y^{m}}{a n +b m}\right ) {\mathrm e}^{-\frac {c \,x^{n} y^{m}}{2 a n +2 b m}}+\left (\left (k +n \right ) a +\left (m +s \right ) b \right ) \left (\left (\left (k +n \right ) a +\left (m +s \right ) b \right ) \left (a n +b m \right ) p \,x^{\frac {b s +\left (k -n \right ) a}{a}} x^{-\frac {b s}{a}} y^{s} y^{-m} \left (\frac {c \,x^{n} y^{m}}{a n +b m}\right )^{\frac {\left (-k -n \right ) a -\left (m +s \right ) b}{2 a n +2 b m}} \left (\frac {c \,x^{-\frac {b m}{a}} y^{m}}{a n +b m}\right )^{\frac {-k a -b s}{a n +b m}} \left (\frac {c \,x^{-\frac {b m}{a}} y^{m}}{a n +b m}\right )^{\frac {k a +b s}{a n +b m}} \WhittakerM \left (\frac {\left (k +n \right ) a +\left (m +s \right ) b}{2 a n +2 b m}, \frac {\left (k +2 n \right ) a +2 \left (m +\frac {s}{2}\right ) b}{2 a n +2 b m}, \frac {c \,x^{n} y^{m}}{a n +b m}\right ) {\mathrm e}^{-\frac {c \,x^{n} y^{m}}{2 a n +2 b m}}+\left (k a +b s \right ) \left (\left (k +2 n \right ) a +2 \left (m +\frac {s}{2}\right ) b \right ) c \textit {\_F1} \left (y \,x^{-\frac {b}{a}}\right )\right )\right ) {\mathrm e}^{\frac {c \,x^{n} y^{m}}{a n +b m}}}{\left (\left (k +n \right ) a +\left (m +s \right ) b \right ) \left (k a +b s \right ) \left (\left (k +2 n \right ) a +2 \left (m +\frac {s}{2}\right ) b \right ) c}\]
____________________________________________________________________________________
Added April 1, 2019.
Problem Chapter 5.2.4.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x w_x + b y w_y = (c x^n+ p y^m) w + q x^k y^s \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == c*(x^n+p*y^m)*w[x,y] + q*x^k*y^s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x^n}{a n}+\frac {c p y^m}{b m}} \left (\int _1^x\frac {\exp \left (-\frac {c \left (\frac {a p \left (x^{-\frac {b}{a}} y K[1]^{\frac {b}{a}}\right )^m}{b m}+\frac {K[1]^n}{n}\right )}{a}\right ) q K[1]^{k-1} \left (x^{-\frac {b}{a}} y K[1]^{\frac {b}{a}}\right )^s}{a}dK[1]+c_1\left (y x^{-\frac {b}{a}}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*x*diff(w(x,y),x)+ b*y*diff(w(x,y),y) = c*(x^n+y^m)*w(x,y)+ q*x^k*y^s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {q \,\textit {\_b}^{k -1} \left (y \,\textit {\_b}^{\frac {b}{a}} x^{-\frac {b}{a}}\right )^{s} {\mathrm e}^{-\frac {c \left (\int \frac {\textit {\_b}^{n}+\left (y \,\textit {\_b}^{\frac {b}{a}} x^{-\frac {b}{a}}\right )^{m}}{\textit {\_b}}d \textit {\_b} \right )}{a}}}{a}d \textit {\_b} +\textit {\_F1} \left (y \,x^{-\frac {b}{a}}\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {\left (\textit {\_a}^{n}+\left (y \,\textit {\_a}^{\frac {b}{a}} x^{-\frac {b}{a}}\right )^{m}\right ) c}{\textit {\_a} a}d \textit {\_a}}\]
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Added April 1, 2019.
Problem Chapter 5.2.4.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x^2 w_x + a x y w_y = b y^2 w + c x^n y^m \]
Mathematica ✓
ClearAll["Global`*"]; pde = x^2*D[w[x, y], x] + a*x*y*D[w[x, y], y] == b*y^2*w[x,y] + c*x^n*y^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{-\frac {b y^2}{x-2 a x}} \left (-\frac {c y^m x^{n-1} \left (-\frac {b y^2}{x-2 a x}\right )^{-\frac {a m+n-1}{2 a-1}} \operatorname {Gamma}\left (\frac {a m+n-1}{2 a-1},-\frac {b y^2}{x-2 a x}\right )}{2 a-1}+c_1\left (y x^{-a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := x^2*diff(w(x,y),x)+ a*x*y*diff(w(x,y),y) = b*y^2*w(x,y)+ c*x^n*y^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \frac {4 \left (\frac {\left (\left (m +2\right ) a +n -2\right )^{2} \left (a -\frac {1}{2}\right ) c x \,x^{-a m} x^{a m +n} y^{m} \left (\frac {b \,y^{2} x^{-2 a}}{2 a -1}\right )^{\frac {-a m -n +1}{2 a -1}} \left (\frac {b \,y^{2} x^{-2 a}}{2 a -1}\right )^{\frac {a m +n -1}{2 a -1}} \left (\frac {b \,y^{2}}{\left (2 a -1\right ) x}\right )^{\frac {\left (-m -2\right ) a -n +2}{4 a -2}} \WhittakerM \left (\frac {\left (m +2\right ) a +n -2}{4 a -2}, \frac {\left (m +4\right ) a +n -3}{4 a -2}, \frac {b \,y^{2}}{\left (2 a -1\right ) x}\right ) {\mathrm e}^{-\frac {b \,y^{2}}{2 \left (2 a -1\right ) x}}}{2}+\left (b \,y^{2}+\left (\left (m +2\right ) a +n -2\right ) x \right ) \left (a -\frac {1}{2}\right )^{2} c \,x^{-a m} x^{a m +n} y^{m} \left (\frac {b \,y^{2} x^{-2 a}}{2 a -1}\right )^{\frac {-a m -n +1}{2 a -1}} \left (\frac {b \,y^{2} x^{-2 a}}{2 a -1}\right )^{\frac {a m +n -1}{2 a -1}} \left (\frac {b \,y^{2}}{\left (2 a -1\right ) x}\right )^{\frac {\left (-m -2\right ) a -n +2}{4 a -2}} \WhittakerM \left (\frac {\left (m -2\right ) a +n}{4 a -2}, \frac {\left (m +4\right ) a +n -3}{4 a -2}, \frac {b \,y^{2}}{\left (2 a -1\right ) x}\right ) {\mathrm e}^{-\frac {b \,y^{2}}{2 \left (2 a -1\right ) x}}+\frac {\left (\left (m +4\right ) a +n -3\right ) \left (a m +n -1\right ) \left (\left (m +2\right ) a +n -2\right ) b x \,y^{2} \textit {\_F1} \left (y \,x^{-a}\right )}{4}\right ) {\mathrm e}^{\frac {b \,y^{2}}{\left (2 a -1\right ) x}}}{\left (\left (m +4\right ) a +n -3\right ) \left (a m +n -1\right ) \left (\left (m +2\right ) a +n -2\right ) b x \,y^{2}}\]
____________________________________________________________________________________
Added April 1, 2019.
Problem Chapter 5.2.4.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x^2 w_x + x y w_y = y^2(a x+b y) w + c x^n y^m \]
Mathematica ✓
ClearAll["Global`*"]; pde = x^2*D[w[x, y], x] + x*y*D[w[x, y], y] == y^2*(a*x+b*y)*w[x,y] + c*x^n*y^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to -c 2^{\frac {1}{2} (m+n-3)} y^m x^{n-1} e^{\frac {1}{2} y^2 \left (a+\frac {b y}{x}\right )} \left (y^2 \left (a+\frac {b y}{x}\right )\right )^{\frac {1}{2} (-m-n+1)} \operatorname {Gamma}\left (\frac {1}{2} (m+n-1),\frac {1}{2} y^2 \left (a+\frac {b y}{x}\right )\right )+c_1\left (\frac {y}{x}\right ) e^{\frac {1}{2} y^2 \left (a+\frac {b y}{x}\right )}\right \}\right \}\]
Maple ✓
restart; pde := x^2*diff(w(x,y),x)+ x*y*diff(w(x,y),y) = y^2*(a*x+b*y)*w(x,y)+ c*x^n*y^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \frac {\left (\left (m +n +3\right ) \left (b \,y^{3}+\left (a \,y^{2}+m +n +1\right ) x \right ) c x 2^{-\frac {m}{4}-\frac {n}{4}+\frac {3}{4}} 2^{\frac {m}{2}+\frac {n}{2}+\frac {1}{2}} x^{-m -2} x^{m +n -1} y^{m +2} \left (\left (a +\frac {b y}{x}\right ) y^{2}\right )^{-\frac {m}{4}-\frac {n}{4}-\frac {1}{4}} \left (\frac {\left (a +\frac {b y}{x}\right ) y^{2}}{x^{2}}\right )^{\frac {m}{2}+\frac {n}{2}-\frac {1}{2}} \left (\frac {\left (a +\frac {b y}{x}\right ) y^{2}}{x^{2}}\right )^{-\frac {m}{2}-\frac {n}{2}-\frac {1}{2}} \WhittakerM \left (\frac {m}{4}+\frac {n}{4}+\frac {5}{4}, \frac {m}{4}+\frac {n}{4}+\frac {3}{4}, \frac {\left (a +\frac {b y}{x}\right ) y^{2}}{2}\right ) {\mathrm e}^{-\frac {a \,y^{2}}{4}-\frac {b \,y^{3}}{4 x}}+\left (\left (a x +b y \right )^{2} c \,y^{2} 2^{-\frac {m}{4}-\frac {n}{4}+\frac {3}{4}} 2^{\frac {m}{2}+\frac {n}{2}+\frac {1}{2}} x^{-m -2} x^{m +n -1} y^{m +2} \left (\left (a +\frac {b y}{x}\right ) y^{2}\right )^{-\frac {m}{4}-\frac {n}{4}-\frac {1}{4}} \left (\frac {\left (a +\frac {b y}{x}\right ) y^{2}}{x^{2}}\right )^{\frac {m}{2}+\frac {n}{2}-\frac {1}{2}} \left (\frac {\left (a +\frac {b y}{x}\right ) y^{2}}{x^{2}}\right )^{-\frac {m}{2}-\frac {n}{2}-\frac {1}{2}} \WhittakerM \left (\frac {m}{4}+\frac {n}{4}+\frac {1}{4}, \frac {m}{4}+\frac {n}{4}+\frac {3}{4}, \frac {\left (a +\frac {b y}{x}\right ) y^{2}}{2}\right ) {\mathrm e}^{-\frac {a \,y^{2}}{4}-\frac {b \,y^{3}}{4 x}}+\left (m +n -1\right ) \left (m +n +3\right ) \left (m +n +1\right ) x^{2} \textit {\_F1} \left (\frac {y}{x}\right )\right ) y^{2}\right ) {\mathrm e}^{\frac {a \,y^{2}}{2}+\frac {b \,y^{3}}{2 x}}}{\left (m +n -1\right ) \left (m +n +1\right ) \left (m +n +3\right ) x^{2} y^{2}}\]
____________________________________________________________________________________
Added April 1, 2019.
Problem Chapter 5.2.4.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x^n w_x + b x^m y w_y = c x^p y^q w+s x^\gamma y^\delta + d \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^n*D[w[x, y], x] + b*x^m*y*D[w[x, y], y] == c*x^p*y^q*w[x,y] + s*x^gamma*y^delta+d; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (-\frac {c y^q x^{-n+p+1} e^{-\frac {b q x^{m-n+1}}{a m-a n+a}} \left (-\frac {b q x^{m-n+1}}{a m-a n+a}\right )^{\frac {n-p-1}{m-n+1}} \operatorname {Gamma}\left (\frac {-n+p+1}{m-n+1},-\frac {b q x^{m-n+1}}{a m-a n+a}\right )}{a (m-n+1)}\right ) \left (\int _1^x\frac {\exp \left (\frac {c e^{-\frac {b q K[1]^{m-n+1}}{m a-n a+a}} \left (e^{\frac {b \left (K[1]^{m-n+1}-x^{m-n+1}\right )}{m a-n a+a}} y\right )^q \operatorname {Gamma}\left (\frac {-n+p+1}{m-n+1},-\frac {b q K[1]^{m-n+1}}{m a-n a+a}\right ) K[1]^{-n+p+1} \left (-\frac {b q K[1]^{m-n+1}}{m a-n a+a}\right )^{\frac {n-p-1}{m-n+1}}}{a (m-n+1)}\right ) K[1]^{-n} \left (s K[1]^{\gamma } \left (e^{\frac {b \left (K[1]^{m-n+1}-x^{m-n+1}\right )}{m a-n a+a}} y\right )^{\delta }+d\right )}{a}dK[1]+c_1\left (y e^{-\frac {b x^{m-n+1}}{a m-a n+a}}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*x^n*diff(w(x,y),x)+ b*x^m*y*diff(w(x,y),y) = c*x^p*y^q*w(x,y)+ s*x^gamma*y^delta+d; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (s \,\textit {\_b}^{-n +\gamma } \left (y \,{\mathrm e}^{-\frac {\left (-\textit {\_b}^{m -n +1}+x^{m -n +1}\right ) b}{\left (m -n +1\right ) a}}\right )^{\delta }+d \,\textit {\_b}^{-n}\right ) {\mathrm e}^{-\frac {c \left (\int \textit {\_b}^{-n +p} \left (y \,{\mathrm e}^{-\frac {\left (-\textit {\_b}^{m -n +1}+x^{m -n +1}\right ) b}{\left (m -n +1\right ) a}}\right )^{q}d \textit {\_b} \right )}{a}}}{a}d \textit {\_b} +\textit {\_F1} \left (y \,{\mathrm e}^{-\frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}}\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {c \,\textit {\_a}^{-n +p} \left (y \,{\mathrm e}^{-\frac {\left (-\textit {\_a}^{m -n +1}+x^{m -n +1}\right ) b}{\left (m -n +1\right ) a}}\right )^{q}}{a}d \textit {\_a}}\]
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Added April 1, 2019.
Problem Chapter 5.2.4.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x^n w_x + (b x^m y +c x^k ) w_y = s x^p y^q w+d \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^n*D[w[x, y], x] + (b*x^m*y+x*x^k)*D[w[x, y], y] == s*x^p*y^q*w[x,y] + d; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {s \left (b^{-\frac {k+2}{m-n+1}} \exp \left (-\frac {b \left (x^{m-n+1}-K[1]^{m-n+1}\right )}{a (m-n+1)}\right ) (a (m-n+1))^{-\frac {m}{m-n+1}} \left (b^{\frac {n}{m-n+1}} e^{\frac {b x^{m-n+1}}{m a-n a+a}} \operatorname {Gamma}\left (\frac {k-n+2}{m-n+1},\frac {b x^{m-n+1}}{m a-n a+a}\right ) (a (m-n+1))^{\frac {k+1}{m-n+1}}-b^{\frac {n}{m-n+1}} e^{\frac {b x^{m-n+1}}{m a-n a+a}} \operatorname {Gamma}\left (\frac {k-n+2}{m-n+1},\frac {b K[1]^{m-n+1}}{m a-n a+a}\right ) (a (m-n+1))^{\frac {k+1}{m-n+1}}+b^{\frac {k+2}{m-n+1}} y (a (m-n+1))^{\frac {m}{m-n+1}}\right )\right )^q K[1]^{p-n}}{a}dK[1]\right ) \left (\int _1^x\frac {d \exp \left (-\int _1^{K[2]}\frac {s \left (b^{-\frac {k+2}{m-n+1}} \exp \left (-\frac {b \left (x^{m-n+1}-K[1]^{m-n+1}\right )}{a (m-n+1)}\right ) (a (m-n+1))^{-\frac {m}{m-n+1}} \left (b^{\frac {n}{m-n+1}} e^{\frac {b x^{m-n+1}}{m a-n a+a}} \operatorname {Gamma}\left (\frac {k-n+2}{m-n+1},\frac {b x^{m-n+1}}{m a-n a+a}\right ) (a (m-n+1))^{\frac {k+1}{m-n+1}}-b^{\frac {n}{m-n+1}} e^{\frac {b x^{m-n+1}}{m a-n a+a}} \operatorname {Gamma}\left (\frac {k-n+2}{m-n+1},\frac {b K[1]^{m-n+1}}{m a-n a+a}\right ) (a (m-n+1))^{\frac {k+1}{m-n+1}}+b^{\frac {k+2}{m-n+1}} y (a (m-n+1))^{\frac {m}{m-n+1}}\right )\right )^q K[1]^{p-n}}{a}dK[1]\right ) K[2]^{-n}}{a}dK[2]+c_1\left ((a (m-n+1))^{\frac {k-m+1}{m-n+1}} b^{\frac {-k+n-2}{m-n+1}} \operatorname {Gamma}\left (\frac {k-n+2}{m-n+1},\frac {b x^{m-n+1}}{a m-a n+a}\right )+y e^{-\frac {b x^{m-n+1}}{a m-a n+a}}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*x^n*diff(w(x,y),x)+ (b*x^m*y+c*x^k)*diff(w(x,y),y) = s*x^p*y^q*w(x,y)+ d; 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 {d \,\textit {\_b}^{-n} {\mathrm e}^{-\frac {s \left (\int \textit {\_b}^{-n +p} \left (\frac {-\left (m -n +1\right ) \left (k +m -2 n +2\right )^{2} a c \,x^{k -m} \left (\frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}\right )^{\frac {-k -m +2 n -2}{2 m -2 n +2}} \WhittakerM \left (\frac {k +m -2 n +2}{2 m -2 n +2}, \frac {k +2 m -3 n +3}{2 m -2 n +2}, \frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}\right ) {\mathrm e}^{\frac {b \,\textit {\_b}^{m -n +1}}{\left (m -n +1\right ) a}} {\mathrm e}^{-\frac {b \,x^{m -n +1}}{2 \left (m -n +1\right ) a}}+\left (m -n +1\right ) \left (k +m -2 n +2\right )^{2} a c \,\textit {\_b}^{k -m} \left (\frac {b \,\textit {\_b}^{m -n +1}}{\left (m -n +1\right ) a}\right )^{\frac {-k -m +2 n -2}{2 m -2 n +2}} \WhittakerM \left (\frac {k +m -2 n +2}{2 m -2 n +2}, \frac {k +2 m -3 n +3}{2 m -2 n +2}, \frac {b \,\textit {\_b}^{m -n +1}}{\left (m -n +1\right ) a}\right ) {\mathrm e}^{\frac {b \,\textit {\_b}^{m -n +1}}{2 \left (m -n +1\right ) a}}+\left (m -n +1\right )^{2} \left (\left (k +m -2 n +2\right ) a \,\textit {\_b}^{k -m}+b \,\textit {\_b}^{k -n +1}\right ) c \left (\frac {b \,\textit {\_b}^{m -n +1}}{\left (m -n +1\right ) a}\right )^{\frac {-k -m +2 n -2}{2 m -2 n +2}} \WhittakerM \left (\frac {k -m}{2 m -2 n +2}, \frac {k +2 m -3 n +3}{2 m -2 n +2}, \frac {b \,\textit {\_b}^{m -n +1}}{\left (m -n +1\right ) a}\right ) {\mathrm e}^{\frac {b \,\textit {\_b}^{m -n +1}}{2 \left (m -n +1\right ) a}}-\left (-\left (k -n +1\right ) \left (k +m -2 n +2\right ) \left (k +2 m -3 n +3\right ) a b y \,{\mathrm e}^{-\frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}}+\left (m -n +1\right )^{2} \left (\left (k +m -2 n +2\right ) a \,x^{k -m}+b \,x^{k -n +1}\right ) c \left (\frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}\right )^{\frac {-k -m +2 n -2}{2 m -2 n +2}} \WhittakerM \left (\frac {k -m}{2 m -2 n +2}, \frac {k +2 m -3 n +3}{2 m -2 n +2}, \frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}\right ) {\mathrm e}^{-\frac {b \,x^{m -n +1}}{2 \left (m -n +1\right ) a}}\right ) {\mathrm e}^{\frac {b \,\textit {\_b}^{m -n +1}}{\left (m -n +1\right ) a}}}{\left (k +2 m -3 n +3\right ) \left (k +m -2 n +2\right ) \left (k -n +1\right ) a b}\right )^{q}d \textit {\_b} \right )}{a}}}{a}d \textit {\_b} +\textit {\_F1} \left (\frac {-\left (m -n +1\right ) \left (k +m -2 n +2\right )^{2} a c \,x^{k -m} \left (\frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}\right )^{\frac {-k -m +2 n -2}{2 m -2 n +2}} \WhittakerM \left (\frac {k +m -2 n +2}{2 m -2 n +2}, \frac {k +2 m -3 n +3}{2 m -2 n +2}, \frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}\right ) {\mathrm e}^{-\frac {b \,x^{m -n +1}}{2 \left (m -n +1\right ) a}}+\left (k -n +1\right ) \left (k +m -2 n +2\right ) \left (k +2 m -3 n +3\right ) a b y \,{\mathrm e}^{-\frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}}-\left (m -n +1\right )^{2} \left (\left (k +m -2 n +2\right ) a \,x^{k -m}+b \,x^{k -n +1}\right ) c \left (\frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}\right )^{\frac {-k -m +2 n -2}{2 m -2 n +2}} \WhittakerM \left (\frac {k -m}{2 m -2 n +2}, \frac {k +2 m -3 n +3}{2 m -2 n +2}, \frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}\right ) {\mathrm e}^{-\frac {b \,x^{m -n +1}}{2 \left (m -n +1\right ) a}}}{\left (k +2 m -3 n +3\right ) \left (k +m -2 n +2\right ) \left (k -n +1\right ) a b}\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {s \,\textit {\_a}^{-n +p} \left (\frac {-\left (m -n +1\right ) \left (k +m -2 n +2\right )^{2} a c \,x^{k -m} \left (\frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}\right )^{\frac {-k -m +2 n -2}{2 m -2 n +2}} \WhittakerM \left (\frac {k +m -2 n +2}{2 m -2 n +2}, \frac {k +2 m -3 n +3}{2 m -2 n +2}, \frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}\right ) {\mathrm e}^{\frac {b \,\textit {\_a}^{m -n +1}}{\left (m -n +1\right ) a}} {\mathrm e}^{-\frac {b \,x^{m -n +1}}{2 \left (m -n +1\right ) a}}+\left (m -n +1\right ) \left (k +m -2 n +2\right )^{2} a c \,\textit {\_a}^{k -m} \left (\frac {b \,\textit {\_a}^{m -n +1}}{\left (m -n +1\right ) a}\right )^{\frac {-k -m +2 n -2}{2 m -2 n +2}} \WhittakerM \left (\frac {k +m -2 n +2}{2 m -2 n +2}, \frac {k +2 m -3 n +3}{2 m -2 n +2}, \frac {b \,\textit {\_a}^{m -n +1}}{\left (m -n +1\right ) a}\right ) {\mathrm e}^{\frac {b \,\textit {\_a}^{m -n +1}}{2 \left (m -n +1\right ) a}}+\left (m -n +1\right )^{2} \left (\left (k +m -2 n +2\right ) a \,\textit {\_a}^{k -m}+b \,\textit {\_a}^{k -n +1}\right ) c \left (\frac {b \,\textit {\_a}^{m -n +1}}{\left (m -n +1\right ) a}\right )^{\frac {-k -m +2 n -2}{2 m -2 n +2}} \WhittakerM \left (\frac {k -m}{2 m -2 n +2}, \frac {k +2 m -3 n +3}{2 m -2 n +2}, \frac {b \,\textit {\_a}^{m -n +1}}{\left (m -n +1\right ) a}\right ) {\mathrm e}^{\frac {b \,\textit {\_a}^{m -n +1}}{2 \left (m -n +1\right ) a}}-\left (-\left (k -n +1\right ) \left (k +m -2 n +2\right ) \left (k +2 m -3 n +3\right ) a b y \,{\mathrm e}^{-\frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}}+\left (m -n +1\right )^{2} \left (\left (k +m -2 n +2\right ) a \,x^{k -m}+b \,x^{k -n +1}\right ) c \left (\frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}\right )^{\frac {-k -m +2 n -2}{2 m -2 n +2}} \WhittakerM \left (\frac {k -m}{2 m -2 n +2}, \frac {k +2 m -3 n +3}{2 m -2 n +2}, \frac {b \,x^{m -n +1}}{\left (m -n +1\right ) a}\right ) {\mathrm e}^{-\frac {b \,x^{m -n +1}}{2 \left (m -n +1\right ) a}}\right ) {\mathrm e}^{\frac {b \,\textit {\_a}^{m -n +1}}{\left (m -n +1\right ) a}}}{\left (k +2 m -3 n +3\right ) \left (k +m -2 n +2\right ) \left (k -n +1\right ) a b}\right )^{q}}{a}d \textit {\_a}}\]
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Added April 1, 2019.
Problem Chapter 5.2.4.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x^n w_x + b x^m y^k w_y = c w + s x^p y^q + d \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^n*D[w[x, y], x] + b*x^m*y^k*D[w[x, y], y] == c*w[x,y] + s*x^p*y^q+d; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x^{1-n}}{a-a n}} \left (\int _1^x\frac {e^{\frac {c K[1]^{1-n}}{a (n-1)}} K[1]^{-n} \left (s \left (\left (-\frac {a (m-n+1) x^n y^k K[1]^n}{a (-m+n-1) x^n y K[1]^n-b (k-1) y^k \left (x^{m+1} K[1]^n-x^n K[1]^{m+1}\right )}\right )^{\frac {1}{k-1}}\right )^q K[1]^p+d\right )}{a}dK[1]+c_1\left (\frac {\frac {y^{1-k} (-m+n-1)}{k-1}-\frac {b x^{m-n+1}}{a}}{m-n+1}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*x^n*diff(w(x,y),x)+ b*x^m*y^k*diff(w(x,y),y) = c*w(x,y)+ s*x^p*y^q+d; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (s \,\textit {\_a}^{-n +p} \left (\left (\frac {\left (m -n +1\right ) a \,y^{-k +1}-\left (k -1\right ) b \,\textit {\_a}^{m -n +1}+\left (k -1\right ) b \,x^{m -n +1}}{\left (m -n +1\right ) a}\right )^{-\frac {1}{k -1}}\right )^{q}+d \,\textit {\_a}^{-n}\right ) {\mathrm e}^{\frac {c \,\textit {\_a}^{-n +1}}{\left (n -1\right ) a}}}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {\left (m -n +1\right ) a \,y^{-k +1}+\left (k -1\right ) b \,x^{m -n +1}}{\left (m -n +1\right ) a}\right )\right ) {\mathrm e}^{-\frac {c \,x^{-n +1}}{\left (n -1\right ) a}}\]
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Added April 1, 2019.
Problem Chapter 5.2.4.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a y^k w_x + b x^n w_y = c w + s x^m \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*y^k*D[w[x, y], x] + b*x^n*D[w[x, y], y] == c*w[x,y] + s*x^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\frac {c x \left (\left (y^{-k-1}\right )^{-\frac {1}{k+1}}\right )^{-k} \left (\frac {a (n+1) y^{k+1}}{a (n+1) y^{k+1}-b (k+1) x^{n+1}}\right )^{\frac {k}{k+1}} \, _2F_1\left (\frac {k}{k+1},\frac {1}{n+1};1+\frac {1}{n+1};\frac {b (k+1) x^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )}{a}\right ) \left (\int _1^x\frac {\exp \left (-\frac {c \, _2F_1\left (\frac {k}{k+1},\frac {1}{n+1};1+\frac {1}{n+1};\frac {b (k+1) K[1]^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right ) K[1] \left (1-\frac {b (k+1) K[1]^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )^{\frac {k}{k+1}} \left (\left (\frac {a (n+1)}{a (n+1) y^{k+1}-b (k+1) \left (x^{n+1}-K[1]^{n+1}\right )}\right )^{-\frac {1}{k+1}}\right )^{-k}}{a}\right ) s K[1]^m \left (\left (\frac {a (n+1)}{a (n+1) y^{k+1}-b (k+1) \left (x^{n+1}-K[1]^{n+1}\right )}\right )^{-\frac {1}{k+1}}\right )^{-k}}{a}dK[1]+c_1\left (\frac {y^{k+1}}{k+1}-\frac {b x^{n+1}}{a n+a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*y^k*diff(w(x,y),x)+ b*x^n*diff(w(x,y),y) = c*w(x,y)+ s*x^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {s \,\textit {\_b}^{m} \left (\left (\frac {\left (n +1\right ) a \,y^{k +1}+\left (k +1\right ) b \,\textit {\_b}^{n +1}-\left (k +1\right ) b \,x^{n +1}}{\left (n +1\right ) a}\right )^{\frac {1}{k +1}}\right )^{-k} {\mathrm e}^{-\frac {c \left (\int \left (\left (\frac {\left (n +1\right ) a \,y^{k +1}+\left (k +1\right ) b \,\textit {\_b}^{n +1}-\left (k +1\right ) b \,x^{n +1}}{\left (n +1\right ) a}\right )^{\frac {1}{k +1}}\right )^{-k}d \textit {\_b} \right )}{a}}}{a}d \textit {\_b} +\textit {\_F1} \left (\frac {\left (n +1\right ) a \,y^{k +1}-\left (k +1\right ) b \,x^{n +1}}{\left (n +1\right ) a}\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {c \left (\left (\frac {\left (n +1\right ) a \,y^{k +1}+\left (k +1\right ) b \,\textit {\_a}^{n +1}-\left (k +1\right ) b \,x^{n +1}}{\left (n +1\right ) a}\right )^{\frac {1}{k +1}}\right )^{-k}}{a}d \textit {\_a}}\]
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