7.3.5 2.4

7.3.5.1 [868] Problem 1

problem number 868

Added Feb. 9, 2019.

Problem Chapter 3.2.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x +b w_y = c x^n + d y^m \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == c*x^n + d*y^m; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right )+\frac {c x^{n+1}}{a n+a}+\frac {d y^{m+1}}{b m+b}\right \}\right \}\]

Maple ✓

restart; 
pde := a*diff(w(x,y),x) + b*diff(w(x,y),y) =c*x^n+d*y^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = \frac {\left (m +1\right ) \left (n +1\right ) a b \textit {\_F1} \left (\frac {a y -b x}{a}\right )+\left (n +1\right ) a d \,y^{m +1}+\left (m +1\right ) b c \,x^{n +1}}{\left (m +1\right ) \left (n +1\right ) a b}\]

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7.3.5.2 [869] Problem 2

problem number 869

Added Feb. 9, 2019.

Problem Chapter 3.2.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x +b w_y = c x^n y \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == c*x^n*y; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \frac {-c x^{n+1} (b x-a (n+2) y)+a^2 \left (n^2+3 n+2\right ) c_1\left (y-\frac {b x}{a}\right )}{a^2 (n+1) (n+2)}\right \}\right \}\]

Maple ✓

restart; 
pde := a*diff(w(x,y),x) + b*diff(w(x,y),y) =c*x^n*y; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = \frac {\left (n +2\right ) \left (n +1\right ) a^{2} \textit {\_F1} \left (\frac {a y -b x}{a}\right )+\left (\left (n +2\right ) a y -b x \right ) c \,x^{n +1}}{\left (n +2\right ) \left (n +1\right ) a^{2}}\]

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7.3.5.3 [870] Problem 3

problem number 870

Added Feb. 9, 2019.

Problem Chapter 3.2.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x +y w_y = a(x^2+y^2)^k \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + y*D[w[x, y], y] == a*(x^2 + y^2)^k; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \frac {a \left (x^2+y^2\right )^k}{2 k}+c_1\left (\frac {y}{x}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=x*diff(w(x,y),x) + y*diff(w(x,y),y) =a*(x^2+y^2)^k; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = \frac {a \left (x^{2}+y^{2}\right )^{k}+2 k \textit {\_F1} \left (\frac {y}{x}\right )}{2 k}\]

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7.3.5.4 [871] Problem 4

problem number 871

Added Feb. 9, 2019.

Problem Chapter 3.2.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a x w_x +b y w_y = c x^n y^m \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == c*x^n*y^m; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \frac {c y^m x^n}{a n+b m}+c_1\left (y x^{-\frac {b}{a}}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=a*x*diff(w(x,y),x) + b*y*diff(w(x,y),y) =c*x^n*y^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = \frac {c \,x^{n} y^{m}}{n a +b m}+\textit {\_F1} \left (y \,x^{-\frac {b}{a}}\right )\]

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7.3.5.5 [872] Problem 5

problem number 872

Added Feb. 9, 2019.

Problem Chapter 3.2.4.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a x w_x +b y w_y = c x^n + d y^m \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == c*x^n + d*y^m; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y x^{-\frac {b}{a}}\right )+\frac {c x^n}{a n}+\frac {d y^m}{b m}\right \}\right \}\]

Maple ✓

restart; 
pde :=a*x*diff(w(x,y),x) + b*y*diff(w(x,y),y) =c*x^n + d*y^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = \int _{}^{x}\frac {c \,\textit {\_a}^{n}+d \left (y \,\textit {\_a}^{\frac {b}{a}} x^{-\frac {b}{a}}\right )^{m}}{\textit {\_a} a}d \textit {\_a} +\textit {\_F1} \left (y \,x^{-\frac {b}{a}}\right )\] Result has unresolved integral

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7.3.5.6 [873] Problem 6

problem number 873

Added Feb. 9, 2019.

Problem Chapter 3.2.4.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ m x w_x +n y w_y = (a x^n+ b y^m)^k \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  m*x*D[w[x, y], x] + n*y*D[w[x, y], y] == (a*x^n + b*y^m)^k; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \frac {\left (a x^n+b y^m\right )^k}{k m n}+c_1\left (y x^{-\frac {n}{m}}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=m*x*diff(w(x,y),x) + n*y*diff(w(x,y),y) =(a*x^n+b*y^m)^k; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = \frac {k m n \textit {\_F1} \left (y \,x^{-\frac {n}{m}}\right )+\left (a \,x^{n}+b \,y^{m}\right )^{k}}{k m n}\] Result has unresolved integral

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7.3.5.7 [874] Problem 7

problem number 874

Added Feb. 9, 2019.

Problem Chapter 3.2.4.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a x^n w_x +b y^m w_y = c x^k+ d y^s \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x^n*D[w[x, y], x] + b*y^m*D[w[x, y], y] == c*x^k + d*y^s; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {b x^{1-n}}{a (n-1)}-\frac {y^{1-m}}{m-1}\right )+\frac {c x^{k-n+1}}{a k-a n+a}-\frac {d y^{1-m} \left (\left (y^{m-1}\right )^{\frac {1}{m-1}}\right )^s}{b (m-s-1)}\right \}\right \}\]

Maple ✓

restart; 
pde :=a*x^n*diff(w(x,y),x) + n*y^m*diff(w(x,y),y) =c*x^k+d*y^s; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = -\frac {\left (k -n +1\right ) a^{2} d \,a^{\frac {s}{m -1}-1} y^{-m +1} \left (\left (n -1\right ) a \,y^{-m +1}\right )^{-\frac {s}{m -1}} \left (n -1\right )^{\frac {s}{m -1}} {\mathrm e}^{-\frac {i \left (\mathrm {csgn}\left (\frac {i}{n -1}\right ) \mathrm {csgn}\left (i a \,y^{-m +1}\right )^{2}-\mathrm {csgn}\left (i a \,y^{-m +1}\right )^{3}-\left (-\mathrm {csgn}\left (\frac {i}{a}\right )+\mathrm {csgn}\left (i y^{-m +1}\right )\right ) \mathrm {csgn}\left (i y^{-m +1}\right )^{2}+\left (-\mathrm {csgn}\left (\frac {i}{a}\right )+\mathrm {csgn}\left (i y^{-m +1}\right )\right ) \mathrm {csgn}\left (i y^{-m +1}\right ) \mathrm {csgn}\left (i a \,y^{-m +1}\right )+\left (-\mathrm {csgn}\left (\frac {i}{n -1}\right )+\mathrm {csgn}\left (i a \,y^{-m +1}\right )\right ) \mathrm {csgn}\left (i a \,y^{-m +1}\right ) \mathrm {csgn}\left (i \left (n -1\right ) a \,y^{-m +1}\right )\right ) \pi s}{2 m -2}}-\left (m -s -1\right ) \left (\left (k -n +1\right ) a \textit {\_F1} \left (\frac {\left (n -1\right ) a \,y^{-m +1}-\left (m -1\right ) n \,x^{-n +1}}{\left (n -1\right ) a}\right )+c \,x^{k -n +1}\right ) n}{\left (k -n +1\right ) \left (m -s -1\right ) a n}\] Result has unresolved integral

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7.3.5.8 [875] Problem 8

problem number 875

Added Feb. 9, 2019.

Problem Chapter 3.2.4.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a x^n w_x +b x^m y w_y = c x^k y^s + d \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x^n*D[w[x, y], x] + b*x^m*y*D[w[x, y], y] == c*x^k*y^s + d; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to x^{1-n} \left (\frac {d}{a-a n}-\frac {c x^k y^s e^{-\frac {b s x^{m-n+1}}{a m-a n+a}} \left (-\frac {b s x^{m-n+1}}{a m-a n+a}\right )^{\frac {-k+n-1}{m-n+1}} \operatorname {Gamma}\left (\frac {k-n+1}{m-n+1},-\frac {b s x^{m-n+1}}{a m-a n+a}\right )}{a (m-n+1)}\right )+c_1\left (y e^{-\frac {b x^{m-n+1}}{a m-a n+a}}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=a*x^n*diff(w(x,y),x) + n*x^m*y*diff(w(x,y),y) =c*x^k*y^s+d; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = \int _{}^{x}\frac {c \,\textit {\_a}^{k -n} \left (y \,{\mathrm e}^{-\frac {\left (-\textit {\_a}^{m -n +1}+x^{m -n +1}\right ) n}{\left (m -n +1\right ) a}}\right )^{s}+d \,\textit {\_a}^{-n}}{a}d \textit {\_a} +\textit {\_F1} \left (y \,{\mathrm e}^{-\frac {n \,x^{m -n +1}}{\left (m -n +1\right ) a}}\right )\] Result has unresolved integral

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7.3.5.9 [876] Problem 9

problem number 876

Added Feb. 9, 2019.

Problem Chapter 3.2.4.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a x^n w_x +(b x^m y + c x^k) w_y = s x^p y^q + d \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x^n*D[w[x, y], x] + (b*x^m*y + c*x^k)*D[w[x, y], y] == s*x^p*y^q + d; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \int _1^x\frac {K[1]^{-n} \left (s \left (b^{-\frac {k+1}{m-n+1}} \exp \left (-\frac {b \left (x^{m-n+1}-K[1]^{m-n+1}\right )}{a (m-n+1)}\right ) (a (m-n+1))^{-\frac {m}{m-n+1}} \left (b^{\frac {n}{m-n+1}} c e^{\frac {b x^{m-n+1}}{m a-n a+a}} \operatorname {Gamma}\left (\frac {k-n+1}{m-n+1},\frac {b x^{m-n+1}}{m a-n a+a}\right ) (a (m-n+1))^{\frac {k}{m-n+1}}-b^{\frac {n}{m-n+1}} c e^{\frac {b x^{m-n+1}}{m a-n a+a}} \operatorname {Gamma}\left (\frac {k-n+1}{m-n+1},\frac {b K[1]^{m-n+1}}{m a-n a+a}\right ) (a (m-n+1))^{\frac {k}{m-n+1}}+b^{\frac {k+1}{m-n+1}} y (a (m-n+1))^{\frac {m}{m-n+1}}\right )\right )^q K[1]^p+d\right )}{a}dK[1]+c_1\left (c (a (m-n+1))^{\frac {k-m}{m-n+1}} b^{\frac {-k+n-1}{m-n+1}} \operatorname {Gamma}\left (\frac {k-n+1}{m-n+1},\frac {b x^{m-n+1}}{a m-a n+a}\right )+y e^{-\frac {b x^{m-n+1}}{a m-a n+a}}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=a*x^n*diff(w(x,y),x) + n*x^m*y*diff(w(x,y),y) =s*x^p*y^q+d; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = \int _{}^{x}\frac {s \,\textit {\_a}^{-n +p} \left (y \,{\mathrm e}^{-\frac {\left (-\textit {\_a}^{m -n +1}+x^{m -n +1}\right ) n}{\left (m -n +1\right ) a}}\right )^{q}+d \,\textit {\_a}^{-n}}{a}d \textit {\_a} +\textit {\_F1} \left (y \,{\mathrm e}^{-\frac {n \,x^{m -n +1}}{\left (m -n +1\right ) a}}\right )\]

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7.3.5.10 [877] Problem 10

problem number 877

Added Feb. 9, 2019.

Problem Chapter 3.2.4.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a x^n w_x +(b x^m y^k + c x^r y) w_y = s x^p y^q + d \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a*x^n*D[w[x, y], x] + (b*x^m*y^k + c*x^r*y)*D[w[x, y], y] == s*x^p*y^q + d; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=a*x^n*diff(w(x,y),x) +(b*x^m*y^k + c*x^r*y)*diff(w(x,y),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 ) = \int _{}^{x}\frac {s \,\textit {\_a}^{-n +p} \left (\left (\frac {\left (-\left (n -r -1\right ) \left (m -2 n +r +2\right )^{2} a b \,\textit {\_a}^{m -r} y^{\frac {1}{n -r -1}} y^{\frac {r}{n -r -1}} y^{\frac {k n}{n -r -1}} \left (\frac {\left (k -1\right ) c}{\left (n -r -1\right ) a}\right )^{\frac {-m +n -1}{n -r -1}} \left (\frac {\left (k -1\right ) c}{\left (n -r -1\right ) a}\right )^{\frac {m -n +1}{n -r -1}} \left (\frac {\left (k -1\right ) c \,\textit {\_a}^{-n +r +1}}{\left (n -r -1\right ) a}\right )^{\frac {m -2 n +r +2}{2 n -2 r -2}} \WhittakerM \left (\frac {-m +2 n -r -2}{2 n -2 r -2}, \frac {-m +3 n -2 r -3}{2 n -2 r -2}, \frac {\left (k -1\right ) c \,\textit {\_a}^{-n +r +1}}{\left (n -r -1\right ) a}\right ) {\mathrm e}^{\frac {c k \,x^{-n +r +1}}{\left (n -r -1\right ) a}} {\mathrm e}^{-\frac {\left (k -1\right ) c \,\textit {\_a}^{-n +r +1}}{2 \left (n -r -1\right ) a}}+\left (-\left (k -1\right ) c \,\textit {\_a}^{-n +r +1}+\left (m -2 n +r +2\right ) a \right ) \left (n -r -1\right )^{2} b \,\textit {\_a}^{m -r} y^{\frac {1}{n -r -1}} y^{\frac {r}{n -r -1}} y^{\frac {k n}{n -r -1}} \left (\frac {\left (k -1\right ) c}{\left (n -r -1\right ) a}\right )^{\frac {-m +n -1}{n -r -1}} \left (\frac {\left (k -1\right ) c}{\left (n -r -1\right ) a}\right )^{\frac {m -n +1}{n -r -1}} \left (\frac {\left (k -1\right ) c \,\textit {\_a}^{-n +r +1}}{\left (n -r -1\right ) a}\right )^{\frac {m -2 n +r +2}{2 n -2 r -2}} \WhittakerM \left (\frac {-m +r}{2 n -2 r -2}, \frac {-m +3 n -2 r -3}{2 n -2 r -2}, \frac {\left (k -1\right ) c \,\textit {\_a}^{-n +r +1}}{\left (n -r -1\right ) a}\right ) {\mathrm e}^{\frac {c k \,x^{-n +r +1}}{\left (n -r -1\right ) a}} {\mathrm e}^{-\frac {\left (k -1\right ) c \,\textit {\_a}^{-n +r +1}}{2 \left (n -r -1\right ) a}}+\left (m -n +1\right ) \left (\left (k -1\right ) b \,y^{\frac {1}{n -r -1}} y^{\frac {r}{n -r -1}} y^{\frac {k n}{n -r -1}} \left (\int x^{m -n} {\mathrm e}^{-\frac {\left (k -1\right ) c \,x^{-n +r +1}}{\left (n -r -1\right ) a}}d x \right ) {\mathrm e}^{\frac {c k \,x^{-n +r +1}}{\left (n -r -1\right ) a}}+a \,y^{\frac {k}{n -r -1}} y^{\frac {n}{n -r -1}} y^{\frac {k r}{n -r -1}} {\mathrm e}^{\frac {c \,x^{-n +r +1}}{\left (n -r -1\right ) a}}\right ) \left (m -3 n +2 r +3\right ) \left (m -2 n +r +2\right ) c \right ) y^{-\frac {1}{n -r -1}} y^{-\frac {r}{n -r -1}} y^{-\frac {k n}{n -r -1}} {\mathrm e}^{-\frac {c k \,x^{-n +r +1}}{\left (n -r -1\right ) a}}}{\left (m -n +1\right ) \left (m -2 n +r +2\right ) \left (m -3 n +2 r +3\right ) a c}\right )^{-\frac {1}{k -1}} {\mathrm e}^{-\frac {c \,\textit {\_a}^{-n +r +1}}{\left (n -r -1\right ) a}}\right )^{q}+d \,\textit {\_a}^{-n}}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {\left (\left (n -r -1\right ) \left (m -2 n +r +2\right )^{2} a b \,x^{m -r} y^{\frac {1}{n -r -1}} y^{\frac {r}{n -r -1}} y^{\frac {k n}{n -r -1}} \left (\frac {\left (k -1\right ) c}{\left (n -r -1\right ) a}\right )^{\frac {-m +n -1}{n -r -1}} \left (\frac {\left (k -1\right ) c}{\left (n -r -1\right ) a}\right )^{\frac {m -n +1}{n -r -1}} \left (\frac {\left (k -1\right ) c \,x^{-n +r +1}}{\left (n -r -1\right ) a}\right )^{\frac {m -2 n +r +2}{2 n -2 r -2}} \WhittakerM \left (\frac {-m +2 n -r -2}{2 n -2 r -2}, \frac {-m +3 n -2 r -3}{2 n -2 r -2}, \frac {\left (k -1\right ) c \,x^{-n +r +1}}{\left (n -r -1\right ) a}\right ) {\mathrm e}^{\frac {c k \,x^{-n +r +1}}{\left (n -r -1\right ) a}} {\mathrm e}^{-\frac {\left (k -1\right ) c \,x^{-n +r +1}}{2 \left (n -r -1\right ) a}}-\left (-\left (k -1\right ) c \,x^{-n +r +1}+\left (m -2 n +r +2\right ) a \right ) \left (n -r -1\right )^{2} b \,x^{m -r} y^{\frac {1}{n -r -1}} y^{\frac {r}{n -r -1}} y^{\frac {k n}{n -r -1}} \left (\frac {\left (k -1\right ) c}{\left (n -r -1\right ) a}\right )^{\frac {-m +n -1}{n -r -1}} \left (\frac {\left (k -1\right ) c}{\left (n -r -1\right ) a}\right )^{\frac {m -n +1}{n -r -1}} \left (\frac {\left (k -1\right ) c \,x^{-n +r +1}}{\left (n -r -1\right ) a}\right )^{\frac {m -2 n +r +2}{2 n -2 r -2}} \WhittakerM \left (\frac {-m +r}{2 n -2 r -2}, \frac {-m +3 n -2 r -3}{2 n -2 r -2}, \frac {\left (k -1\right ) c \,x^{-n +r +1}}{\left (n -r -1\right ) a}\right ) {\mathrm e}^{\frac {c k \,x^{-n +r +1}}{\left (n -r -1\right ) a}} {\mathrm e}^{-\frac {\left (k -1\right ) c \,x^{-n +r +1}}{2 \left (n -r -1\right ) a}}+\left (m -n +1\right ) \left (m -2 n +r +2\right ) \left (m -3 n +2 r +3\right ) a c \,y^{\frac {k}{n -r -1}} y^{\frac {n}{n -r -1}} y^{\frac {k r}{n -r -1}} {\mathrm e}^{\frac {c \,x^{-n +r +1}}{\left (n -r -1\right ) a}}\right ) y^{-\frac {1}{n -r -1}} y^{-\frac {r}{n -r -1}} y^{-\frac {k n}{n -r -1}} {\mathrm e}^{-\frac {c k \,x^{-n +r +1}}{\left (n -r -1\right ) a}}}{\left (m -n +1\right ) \left (m -2 n +r +2\right ) \left (m -3 n +2 r +3\right ) a c}\right )\]

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7.3.5.11 [878] Problem 11

problem number 878

Added Feb. 9, 2019.

Problem Chapter 3.2.4.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a y^k w_x + b x^m w_y = c x^m + d \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*y^k*D[w[x, y], x] + b*x^m*D[w[x, y], y] == c*x^m + d; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \frac {\left (\left (y^{-k-1}\right )^{-\frac {1}{k+1}}\right )^{-k} \left (b d x \left (\frac {a (m+1) y^{k+1}}{a (m+1) y^{k+1}-b (k+1) x^{m+1}}\right )^{\frac {k}{k+1}} \, _2F_1\left (\frac {k}{k+1},\frac {1}{m+1};1+\frac {1}{m+1};\frac {b (k+1) x^{m+1}}{b (k+1) x^{m+1}-a (m+1) y^{k+1}}\right )+a c y^{k+1}\right )}{a b}+c_1\left (\frac {y^{k+1}}{k+1}-\frac {b x^{m+1}}{a m+a}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=a*y^k*diff(w(x,y),x) +b*x^n*diff(w(x,y),y) =c*x^m+d; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = \int _{}^{x}\frac {\left (c \,\textit {\_a}^{m}+d \right ) \left (\left (\frac {\left (n +1\right ) a \,y^{k +1}+\left (k +1\right ) b \,\textit {\_a}^{n +1}-\left (k +1\right ) b \,x^{n +1}}{\left (n +1\right ) a}\right )^{\frac {1}{k +1}}\right )^{-k}}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {\left (n +1\right ) a \,y^{k +1}-\left (k +1\right ) b \,x^{n +1}}{\left (n +1\right ) a}\right )\]

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