6.5.4 2.4
6.5.4.1 [1219] Problem 1
problem number 1219
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'));
\[w \left (x , y\right ) = \frac {\left (f_{1} \left (\frac {a y -b x}{a}\right ) a +k \int _{}^{x}\textit {\_a}^{n} \left (\frac {a y -b \left (x -\textit {\_a} \right )}{a}\right )^{m} {\mathrm e}^{-\frac {c \textit {\_a}}{a}}d \textit {\_a} \right ) {\mathrm e}^{\frac {c x}{a}}}{a}\]
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6.5.4.2 [1220] Problem 2
problem number 1220
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]];
\[\left \{\left \{w(x,y)\to e^{\frac {x (b-m)}{a}} \left (-\frac {c y^m x^n \left (\frac {x (b-m)}{a}\right )^{-n} \Gamma \left (n+1,\frac {(b-m) x}{a}\right )}{b-m}+e^{\frac {m x}{a}} c_1\left (y e^{-\frac {x}{a}}\right )\right )\right \}\right \}\]
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'));
\[w \left (x , y\right ) = \left (\frac {c \int _{}^{x}\textit {\_a}^{n} \left (y \,{\mathrm e}^{\frac {-x +\textit {\_a}}{a}}\right )^{m} {\mathrm e}^{-\frac {b \textit {\_a}}{a}}d \textit {\_a}}{a}+f_{1} \left (y \,{\mathrm e}^{-\frac {x}{a}}\right )\right ) {\mathrm e}^{\frac {b x}{a}}\]
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6.5.4.3 [1221] Problem 3
problem number 1221
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]];
\[\left \{\left \{w(x,y)\to e^{a x} \left (-b y^m x^n (a x)^{-m-n} \Gamma (m+n,a x)+c_1\left (\frac {y}{x}\right )\right )\right \}\right \}\]
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'));
\[w \left (x , y\right ) = \frac {b \left (a x \right )^{-\frac {n}{2}-\frac {m}{2}} \operatorname {WhittakerM}\left (\frac {n}{2}+\frac {m}{2}, \frac {n}{2}+\frac {m}{2}+\frac {1}{2}, a x \right ) y^{m} x^{n} {\mathrm e}^{\frac {a x}{2}}}{\left (n +m \right ) \left (1+n +m \right )}+\frac {b \,x^{n} y^{m}}{n +m}+{\mathrm e}^{a x} f_{1} \left (\frac {y}{x}\right )\]
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6.5.4.4 [1222] Problem 4
problem number 1222
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'));
\[w \left (x , y\right ) = \frac {y^{m} \operatorname {WhittakerM}\left (\frac {n}{2}+\frac {m}{2}, \frac {n}{2}+\frac {m}{2}+\frac {1}{2}, \sqrt {x^{2}+y^{2}}\, a \right ) \left (\sqrt {x^{2}+y^{2}}\, a \right )^{-\frac {n}{2}-\frac {m}{2}} x^{n} b \,{\mathrm e}^{\frac {\sqrt {x^{2}+y^{2}}\, a}{2}}+\left (1+n +m \right ) \left (f_{1} \left (\frac {y}{x}\right ) \left (n +m \right ) {\mathrm e}^{\sqrt {x^{2}+y^{2}}\, a}+b \,x^{n} y^{m}\right )}{\left (n +m \right ) \left (1+n +m \right )}\]
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6.5.4.5 [1223] Problem 5
problem number 1223
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'));
\[w \left (x , y\right ) = \frac {\left (y^{m} x^{n} c +\left (k +n \right ) a +\left (m +s \right ) b \right ) {\mathrm e}^{\frac {y^{m} x^{n} c}{2 a n +2 b m}} x^{k -n} p \left (a n +b m \right )^{2} \left (\frac {y^{m} x^{n} c}{a n +b m}\right )^{\frac {\left (-k -n \right ) a -\left (m +s \right ) b}{2 a n +2 b m}} y^{-m +s} \operatorname {WhittakerM}\left (\frac {\left (k -n \right ) a -b \left (m -s \right )}{2 a n +2 b m}, \frac {\left (k +2 n \right ) a +2 b \left (m +\frac {s}{2}\right )}{2 a n +2 b m}, \frac {y^{m} x^{n} c}{a n +b m}\right )+\left (\left (k +n \right ) a +\left (m +s \right ) b \right ) \left (\left (\left (k +n \right ) a +\left (m +s \right ) b \right ) {\mathrm e}^{\frac {y^{m} x^{n} c}{2 a n +2 b m}} x^{k -n} p \left (a n +b m \right ) \left (\frac {y^{m} x^{n} c}{a n +b m}\right )^{\frac {\left (-k -n \right ) a -\left (m +s \right ) b}{2 a n +2 b m}} y^{-m +s} \operatorname {WhittakerM}\left (\frac {\left (k +n \right ) a +\left (m +s \right ) b}{2 a n +2 b m}, \frac {\left (k +2 n \right ) a +2 b \left (m +\frac {s}{2}\right )}{2 a n +2 b m}, \frac {y^{m} x^{n} c}{a n +b m}\right )+\left (a k +b s \right ) f_{1} \left (y \,x^{-\frac {b}{a}}\right ) \left (\left (k +2 n \right ) a +2 b \left (m +\frac {s}{2}\right )\right ) c \,{\mathrm e}^{\frac {y^{m} x^{n} c}{a n +b m}}\right )}{\left (a k +b s \right ) \left (\left (k +n \right ) a +\left (m +s \right ) b \right ) \left (\left (k +2 n \right ) a +2 b \left (m +\frac {s}{2}\right )\right ) c}\]
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6.5.4.6 [1224] Problem 6
problem number 1224
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'));
\[w \left (x , y\right ) = \left (\frac {q \int _{}^{x}\frac {\textit {\_a}^{k} \left (y \,x^{-\frac {b}{a}} \textit {\_a}^{\frac {b}{a}}\right )^{s} {\mathrm e}^{-\frac {c \int \frac {\textit {\_a}^{n}+\left (y \,x^{-\frac {b}{a}} \textit {\_a}^{\frac {b}{a}}\right )^{m}}{\textit {\_a}}d \textit {\_a}}{a}}}{\textit {\_a}}d \textit {\_a}}{a}+f_{1} \left (y \,x^{-\frac {b}{a}}\right )\right ) {\mathrm e}^{\frac {c \int _{}^{x}\frac {\textit {\_a}^{n}+\left (y \,x^{-\frac {b}{a}} \textit {\_a}^{\frac {b}{a}}\right )^{m}}{\textit {\_a}}d \textit {\_a}}{a}}\]
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6.5.4.7 [1225] Problem 7
problem number 1225
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]];
\[\left \{\left \{w(x,y)\to e^{-\frac {b y^2}{x-2 a x}} \left (-\frac {c y^m x^{n-1} \left (-\frac {b y^2}{x-2 a x}\right )^{-\frac {a m+n-1}{2 a-1}} \Gamma \left (\frac {a m+n-1}{2 a-1},-\frac {b y^2}{x-2 a x}\right )}{2 a-1}+c_1\left (y x^{-a}\right )\right )\right \}\right \}\]
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'));
\[w \left (x , y\right ) = \frac {2 \left (\left (m +2\right ) a +n -2\right )^{2} c \,x^{n} x \,{\mathrm e}^{\frac {y^{2} b}{\left (4 a -2\right ) x}} \left (-\frac {1}{2}+a \right ) y^{m} \left (\frac {y^{2} b}{\left (-1+2 a \right ) x}\right )^{\frac {\left (-m -2\right ) a -n +2}{4 a -2}} \operatorname {WhittakerM}\left (\frac {\left (m +2\right ) a +n -2}{4 a -2}, \frac {\left (m +4\right ) a +n -3}{4 a -2}, \frac {y^{2} b}{\left (-1+2 a \right ) x}\right )+4 c \,x^{n} \left (\left (\left (m +2\right ) a +n -2\right ) x +y^{2} b \right ) {\mathrm e}^{\frac {y^{2} b}{\left (4 a -2\right ) x}} \left (-\frac {1}{2}+a \right )^{2} y^{m} \left (\frac {y^{2} b}{\left (-1+2 a \right ) x}\right )^{\frac {\left (-m -2\right ) a -n +2}{4 a -2}} \operatorname {WhittakerM}\left (\frac {\left (m -2\right ) a +n}{4 a -2}, \frac {\left (m +4\right ) a +n -3}{4 a -2}, \frac {y^{2} b}{\left (-1+2 a \right ) x}\right )+\left (\left (m +2\right ) a +n -2\right ) f_{1} \left (y \,x^{-a}\right ) {\mathrm e}^{\frac {y^{2} b}{\left (-1+2 a \right ) x}} x \left (\left (m +4\right ) a +n -3\right ) y^{2} b \left (m a +n -1\right )}{\left (\left (m +2\right ) a +n -2\right ) x \left (\left (m +4\right ) a +n -3\right ) y^{2} b \left (m a +n -1\right )}\]
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6.5.4.8 [1226] Problem 8
problem number 1226
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]];
\[\left \{\left \{w(x,y)\to -c 2^{\frac {1}{2} (m+n-3)} y^m x^{n-1} e^{\frac {1}{2} y^2 \left (a+\frac {b y}{x}\right )} \left (y^2 \left (a+\frac {b y}{x}\right )\right )^{\frac {1}{2} (-m-n+1)} \Gamma \left (\frac {1}{2} (m+n-1),\frac {1}{2} y^2 \left (a+\frac {b y}{x}\right )\right )+c_1\left (\frac {y}{x}\right ) e^{\frac {1}{2} y^2 \left (a+\frac {b y}{x}\right )}\right \}\right \}\]
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'));
\[w \left (x , y\right ) = \left (\frac {c 2^{\frac {n}{2}-\frac {1}{2}+\frac {m}{2}} x^{n -2} y^{m} \left (\frac {2^{\frac {7}{4}-\frac {n}{4}-\frac {m}{4}} y^{2} \left (a x +b y \right ) \left (\frac {y^{2} \left (a x +b y \right )}{x}\right )^{-\frac {1}{4}-\frac {n}{4}-\frac {m}{4}} {\mathrm e}^{-\frac {y^{3} b}{4 x}-\frac {y^{2} a}{4}} \operatorname {WhittakerM}\left (\frac {1}{4}+\frac {n}{4}+\frac {m}{4}, \frac {n}{4}+\frac {m}{4}+\frac {3}{4}, \frac {y^{3} b}{2 x}+\frac {y^{2} a}{2}\right )}{n +m +3}+2^{\frac {1}{2}-\frac {n}{2}-\frac {m}{2}} {\mathrm e}^{-\frac {y^{2} a}{2}-\frac {y^{3} b}{2 x}} \left (y^{2} a x +y^{3} b +m x +n x +x \right )\right )}{\left (1+n +m \right ) \left (n -1+m \right )}+f_{1} \left (\frac {y}{x}\right )\right ) {\mathrm e}^{\frac {y^{2} \left (a x +b y \right )}{2 x}}\]
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6.5.4.9 [1227] Problem 9
problem number 1227
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]];
\[\left \{\left \{w(x,y)\to \exp \left (-\frac {c y^q x^{-n+p+1} e^{-\frac {b q x^{m-n+1}}{a m-a n+a}} \left (-\frac {b q x^{m-n+1}}{a m-a n+a}\right )^{\frac {n-p-1}{m-n+1}} \Gamma \left (\frac {-n+p+1}{m-n+1},-\frac {b q x^{m-n+1}}{m a-n a+a}\right )}{a (m-n+1)}\right ) \left (\int _1^x\frac {\exp \left (\frac {c e^{-\frac {b q K[1]^{m-n+1}}{m a-n a+a}} \left (e^{\frac {b \left (K[1]^{m-n+1}-x^{m-n+1}\right )}{m a-n a+a}} y\right )^q \Gamma \left (\frac {-n+p+1}{m-n+1},-\frac {b q K[1]^{m-n+1}}{m a-n a+a}\right ) K[1]^{-n+p+1} \left (-\frac {b q K[1]^{m-n+1}}{m a-n a+a}\right )^{\frac {n-p-1}{m-n+1}}}{a (m-n+1)}\right ) K[1]^{-n} \left (s K[1]^{\gamma } \left (e^{\frac {b \left (K[1]^{m-n+1}-x^{m-n+1}\right )}{m a-n a+a}} y\right )^{\delta }+d\right )}{a}dK[1]+c_1\left (y e^{-\frac {b x^{m-n+1}}{a m-a n+a}}\right )\right )\right \}\right \}\]
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'));
\[w \left (x , y\right ) = \left (\frac {\int _{}^{x}{\mathrm e}^{-\frac {c \int \textit {\_a}^{p -n} \left (y \,{\mathrm e}^{\frac {b \left (-x^{m -n +1}+\textit {\_a}^{m -n +1}\right )}{a \left (m -n +1\right )}}\right )^{q}d \textit {\_a}}{a}} \left (s \,\textit {\_a}^{\gamma -n} \left (y \,{\mathrm e}^{\frac {b \left (-x^{m -n +1}+\textit {\_a}^{m -n +1}\right )}{a \left (m -n +1\right )}}\right )^{\delta }+\textit {\_a}^{-n} d \right )d \textit {\_a}}{a}+f_{1} \left (y \,{\mathrm e}^{-\frac {x^{m -n +1} b}{a \left (m -n +1\right )}}\right )\right ) {\mathrm e}^{\frac {c \int _{}^{x}\textit {\_a}^{p -n} \left (y \,{\mathrm e}^{\frac {b \left (-x^{m -n +1}+\textit {\_a}^{m -n +1}\right )}{a \left (m -n +1\right )}}\right )^{q}d \textit {\_a}}{a}}\]
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6.5.4.10 [1228] Problem 10
problem number 1228
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]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {s \left (b^{-\frac {k+2}{m-n+1}} \exp \left (-\frac {b \left (x^{m-n+1}-K[1]^{m-n+1}\right )}{a (m-n+1)}\right ) (a (m-n+1))^{-\frac {m}{m-n+1}} \left (b^{\frac {n}{m-n+1}} e^{\frac {b x^{m-n+1}}{m a-n a+a}} \Gamma \left (\frac {k-n+2}{m-n+1},\frac {b x^{m-n+1}}{m a-n a+a}\right ) (a (m-n+1))^{\frac {k+1}{m-n+1}}-b^{\frac {n}{m-n+1}} e^{\frac {b x^{m-n+1}}{m a-n a+a}} \Gamma \left (\frac {k-n+2}{m-n+1},\frac {b K[1]^{m-n+1}}{m a-n a+a}\right ) (a (m-n+1))^{\frac {k+1}{m-n+1}}+b^{\frac {k+2}{m-n+1}} y (a (m-n+1))^{\frac {m}{m-n+1}}\right )\right )^q K[1]^{p-n}}{a}dK[1]\right ) \left (\int _1^x\frac {d \exp \left (-\int _1^{K[2]}\frac {s \left (b^{-\frac {k+2}{m-n+1}} \exp \left (-\frac {b \left (x^{m-n+1}-K[1]^{m-n+1}\right )}{a (m-n+1)}\right ) (a (m-n+1))^{-\frac {m}{m-n+1}} \left (b^{\frac {n}{m-n+1}} e^{\frac {b x^{m-n+1}}{m a-n a+a}} \Gamma \left (\frac {k-n+2}{m-n+1},\frac {b x^{m-n+1}}{m a-n a+a}\right ) (a (m-n+1))^{\frac {k+1}{m-n+1}}-b^{\frac {n}{m-n+1}} e^{\frac {b x^{m-n+1}}{m a-n a+a}} \Gamma \left (\frac {k-n+2}{m-n+1},\frac {b K[1]^{m-n+1}}{m a-n a+a}\right ) (a (m-n+1))^{\frac {k+1}{m-n+1}}+b^{\frac {k+2}{m-n+1}} y (a (m-n+1))^{\frac {m}{m-n+1}}\right )\right )^q K[1]^{p-n}}{a}dK[1]\right ) K[2]^{-n}}{a}dK[2]+c_1\left ((a (m-n+1))^{\frac {k-m+1}{m-n+1}} b^{\frac {-k+n-2}{m-n+1}} \Gamma \left (\frac {k-n+2}{m-n+1},\frac {b x^{m-n+1}}{m a-n a+a}\right )+y e^{-\frac {b x^{m-n+1}}{a m-a n+a}}\right )\right )\right \}\right \}\]
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'));
\[\text {Expression too large to display}\]
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6.5.4.11 [1229] Problem 11
problem number 1229
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'));
\[w \left (x , y\right ) = \frac {\left (f_{1} \left (\frac {b \left (k -1\right ) x^{m -n +1}+y^{1-k} \left (m -n +1\right ) a}{\left (m -n +1\right ) a}\right ) a +\int _{}^{x}{\mathrm e}^{\frac {c \,\textit {\_a}^{1-n}}{\left (-1+n \right ) a}} \left (s \,\textit {\_a}^{p -n} {\left (\left (\frac {-b \left (k -1\right ) \textit {\_a}^{m -n +1}+b \left (k -1\right ) x^{m -n +1}+y^{1-k} \left (m -n +1\right ) a}{\left (m -n +1\right ) a}\right )^{-\frac {1}{k -1}}\right )}^{q}+\textit {\_a}^{-n} d \right )d \textit {\_a} \right ) {\mathrm e}^{-\frac {c \,x^{1-n}}{\left (-1+n \right ) a}}}{a}\]
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6.5.4.12 [1230] Problem 12
problem number 1230
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]];
\[\left \{\left \{w(x,y)\to \exp \left (\frac {c x \left (\left (y^{-k-1}\right )^{-\frac {1}{k+1}}\right )^{-k} \left (\frac {a (n+1) y^{k+1}}{a (n+1) y^{k+1}-b (k+1) x^{n+1}}\right )^{\frac {k}{k+1}} \operatorname {Hypergeometric2F1}\left (\frac {k}{k+1},\frac {1}{n+1},1+\frac {1}{n+1},\frac {b (k+1) x^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )}{a}\right ) \left (\int _1^x\frac {\exp \left (-\frac {c \operatorname {Hypergeometric2F1}\left (\frac {k}{k+1},\frac {1}{n+1},1+\frac {1}{n+1},\frac {b (k+1) K[1]^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right ) K[1] \left (1-\frac {b (k+1) K[1]^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )^{\frac {k}{k+1}} \left (\left (\frac {a (n+1)}{a (n+1) y^{k+1}-b (k+1) \left (x^{n+1}-K[1]^{n+1}\right )}\right )^{-\frac {1}{k+1}}\right )^{-k}}{a}\right ) s K[1]^m \left (\left (\frac {a (n+1)}{a (n+1) y^{k+1}-b (k+1) \left (x^{n+1}-K[1]^{n+1}\right )}\right )^{-\frac {1}{k+1}}\right )^{-k}}{a}dK[1]+c_1\left (\frac {y^{k+1}}{k+1}-\frac {b x^{n+1}}{a n+a}\right )\right )\right \}\right \}\]
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'));
\[w \left (x , y\right ) = \frac {\left (f_{1} \left (\frac {-b x \left (1+k \right ) x^{n}+y \,y^{k} a \left (n +1\right )}{\left (n +1\right ) a}\right ) a +s \int _{}^{x}\textit {\_a}^{m} {\left (\left (\frac {\left (1+k \right ) b \,\textit {\_a}^{n +1}-b \,x^{n +1} \left (1+k \right )+y^{1+k} \left (n +1\right ) a}{\left (n +1\right ) a}\right )^{\frac {1}{1+k}}\right )}^{-k} {\mathrm e}^{-\frac {c \int {\left (\left (\frac {\left (1+k \right ) b \,\textit {\_a}^{n +1}-b \,x^{n +1} \left (1+k \right )+y^{1+k} \left (n +1\right ) a}{\left (n +1\right ) a}\right )^{\frac {1}{1+k}}\right )}^{-k}d \textit {\_a}}{a}}d \textit {\_a} \right ) {\mathrm e}^{\frac {c \int _{}^{x}{\left (\left (\frac {\left (1+k \right ) b \,\textit {\_a}^{n +1}-b \,x^{n +1} \left (1+k \right )+y^{1+k} \left (n +1\right ) a}{\left (n +1\right ) a}\right )^{\frac {1}{1+k}}\right )}^{-k}d \textit {\_a}}{a}}}{a}\]
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