____________________________________________________________________________________
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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; 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]];
\[ \left \{\left \{w(x,y)\to \frac {a b m n c_1\left (\frac {a y-b x}{a}\right )+a b m c_1\left (\frac {a y-b x}{a}\right )+a b n c_1\left (\frac {a y-b x}{a}\right )+a b c_1\left (\frac {a y-b x}{a}\right )+a d n y \left (\frac {a y-b x}{a}+\frac {b x}{a}\right )^m+a d y \left (\frac {a y-b x}{a}+\frac {b x}{a}\right )^m+b c m x^{n+1}+b c x^{n+1}}{a b (m+1) (n+1)}\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 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'));
\[ w \left ( x,y \right ) ={\frac {d}{b \left ( m+1 \right ) } \left ( {\frac {bx}{a}}+{\frac {ya-bx}{a}} \right ) ^{m+1}}+{\frac {{x}^{n+1}c}{a \left ( n+1 \right ) }}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \]
____________________________________________________________________________________
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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; 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]];
\[ \left \{\left \{w(x,y)\to \frac {a^2 n^2 c_1\left (\frac {a y-b x}{a}\right )+3 a^2 n c_1\left (\frac {a y-b x}{a}\right )+2 a^2 c_1\left (\frac {a y-b x}{a}\right )+2 a c y x^{n+1}+a c n y x^{n+1}-b c x^{n+2}}{a^2 (n+1) (n+2)}\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 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'));
\[ w \left ( x,y \right ) =-{\frac {{x}^{n+1}bcx}{ \left ( n+2 \right ) \left ( n+1 \right ) {a}^{2}}}+{\frac {{x}^{n+1} \left ( an+2\,a \right ) cy}{ \left ( n+2 \right ) \left ( n+1 \right ) {a}^{2}}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \]
____________________________________________________________________________________
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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; 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]];
\[ \left \{\left \{w(x,y)\to \frac {a \left (x^2 \left (\frac {y^2}{x^2}+1\right )\right )^k+2 k c_1\left (\frac {y}{x}\right )}{2 k}\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 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'));
\[ w \left ( x,y \right ) =1/2\,{\frac {1}{k} \left ( a \left ( {x}^{2} \left ( {\frac {{y}^{2}}{{x}^{2}}}+1 \right ) \right ) ^{k}+2\,{\it \_F1} \left ( {\frac {y}{x}} \right ) k \right ) } \]
____________________________________________________________________________________
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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; 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]];
\[ \left \{\left \{w(x,y)\to \frac {b m c_1\left (y x^{-\frac {b}{a}}\right )+a n c_1\left (y x^{-\frac {b}{a}}\right )+c y^m x^n}{a n+b m}\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 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'));
\[ w \left ( x,y \right ) ={\frac {c{x}^{n}{y}^{m}}{an+bm}}+{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) \]
____________________________________________________________________________________
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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; 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]];
\[ \left \{\left \{w(x,y)\to \frac {a b m n c_1\left (y x^{-\frac {b}{a}}\right )+a d n y^m+b c m x^n}{a b m n}\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 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'));
\[ w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{a{\it \_a}} \left ( {{\it \_a}}^{n}c+d \left ( y{x}^{-{\frac {b}{a}}}{{\it \_a}}^{{\frac {b}{a}}} \right ) ^{m} \right ) }{d{\it \_a}}+{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) \] Result has unresolved integral
____________________________________________________________________________________
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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; 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]];
\[ \left \{\left \{w(x,y)\to \frac {\left (a x^n+b y^m\right )^k+k m n c_1\left (y x^{-\frac {n}{m}}\right )}{k m n}\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 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'));
\[ w \left ( x,y \right ) ={\frac {1}{knm} \left ( {\it \_F1} \left ( y{x}^{-{\frac {n}{m}}} \right ) knm+ \left ( {x}^{n}a+{y}^{m}b \right ) ^{k} \right ) } \] Result has unresolved integral
____________________________________________________________________________________
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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; 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 \frac {x^{-n} \left (-b c x^{k+1}+b c m x^{k+1}-b c s x^{k+1}-a d \left (\left (\frac {a (n-1) x^n}{(m-1) \left (\frac {\left (b x y^m-b m x y^m-a x^n y+a n x^n y\right ) y^{-m}}{m-1}+b x\right )}\right )^{\frac {1}{m-1}}\right )^{s+1} \left (\frac {a (n-1) x^n}{(m-1) \left (\frac {\left (b x y^m-b m x y^m-a x^n y+a n x^n y\right ) y^{-m}}{m-1}+b x\right )}\right )^{\frac {m}{1-m}} x^n-a d k \left (\left (\frac {a (n-1) x^n}{(m-1) \left (\frac {\left (b x y^m-b m x y^m-a x^n y+a n x^n y\right ) y^{-m}}{m-1}+b x\right )}\right )^{\frac {1}{m-1}}\right )^{s+1} \left (\frac {a (n-1) x^n}{(m-1) \left (\frac {\left (b x y^m-b m x y^m-a x^n y+a n x^n y\right ) y^{-m}}{m-1}+b x\right )}\right )^{\frac {m}{1-m}} x^n+a d n \left (\left (\frac {a (n-1) x^n}{(m-1) \left (\frac {\left (b x y^m-b m x y^m-a x^n y+a n x^n y\right ) y^{-m}}{m-1}+b x\right )}\right )^{\frac {1}{m-1}}\right )^{s+1} \left (\frac {a (n-1) x^n}{(m-1) \left (\frac {\left (b x y^m-b m x y^m-a x^n y+a n x^n y\right ) y^{-m}}{m-1}+b x\right )}\right )^{\frac {m}{1-m}} x^n-a b c_1\left (-\frac {x^{-n} y^{-m} \left (b x y^m-b m x y^m-a x^n y+a n x^n y\right )}{a (m-1) (n-1)}\right ) x^n-a b k c_1\left (-\frac {x^{-n} y^{-m} \left (b x y^m-b m x y^m-a x^n y+a n x^n y\right )}{a (m-1) (n-1)}\right ) x^n+a b m c_1\left (-\frac {x^{-n} y^{-m} \left (b x y^m-b m x y^m-a x^n y+a n x^n y\right )}{a (m-1) (n-1)}\right ) x^n+a b k m c_1\left (-\frac {x^{-n} y^{-m} \left (b x y^m-b m x y^m-a x^n y+a n x^n y\right )}{a (m-1) (n-1)}\right ) x^n+a b n c_1\left (-\frac {x^{-n} y^{-m} \left (b x y^m-b m x y^m-a x^n y+a n x^n y\right )}{a (m-1) (n-1)}\right ) x^n-a b m n c_1\left (-\frac {x^{-n} y^{-m} \left (b x y^m-b m x y^m-a x^n y+a n x^n y\right )}{a (m-1) (n-1)}\right ) x^n-a b s c_1\left (-\frac {x^{-n} y^{-m} \left (b x y^m-b m x y^m-a x^n y+a n x^n y\right )}{a (m-1) (n-1)}\right ) x^n-a b k s c_1\left (-\frac {x^{-n} y^{-m} \left (b x y^m-b m x y^m-a x^n y+a n x^n y\right )}{a (m-1) (n-1)}\right ) x^n+a b n s c_1\left (-\frac {x^{-n} y^{-m} \left (b x y^m-b m x y^m-a x^n y+a n x^n y\right )}{a (m-1) (n-1)}\right ) x^n\right )}{a b (k-n+1) (m-s-1)}\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 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 {c{x}^{k-n+1}}{a \left ( k-n+1 \right ) }}-{\frac {d}{ \left ( mn-ns-m-n+s+1 \right ) n} \left ( n-1 \right ) ^{{\frac {s}{m-1}}} \left ( {x}^{-n+1}nm-{\frac {n \left ( {x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a \right ) }{n-1}}-{x}^{-n+1}n+{\frac {{x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a}{n-1}} \right ) \left ( \left ( -{\frac {n \left ( {x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a \right ) }{a \left ( n-1 \right ) }}+{\frac {{x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a}{a \left ( n-1 \right ) }} \right ) a+{x}^{-n+1}nm-{x}^{-n+1}n \right ) ^{-{\frac {s}{m-1}}}{a}^{{\frac {s}{m-1}}-1}{{\rm e}^{{\frac {i/2s\pi }{m-1} \left ( {\it csgn} \left ( {\frac {i}{n-1} \left ( \left ( -{\frac {n \left ( {x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a \right ) }{a \left ( n-1 \right ) }}+{\frac {{x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a}{a \left ( n-1 \right ) }} \right ) a+{x}^{-n+1}nm-{x}^{-n+1}n \right ) } \right ) \right ) ^{3}}-{\frac {i/2s\pi }{m-1} \left ( {\it csgn} \left ( {\frac {i}{n-1} \left ( \left ( -{\frac {n \left ( {x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a \right ) }{a \left ( n-1 \right ) }}+{\frac {{x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a}{a \left ( n-1 \right ) }} \right ) a+{x}^{-n+1}nm-{x}^{-n+1}n \right ) } \right ) \right ) ^{2}{\it csgn} \left ( {\frac {i}{n-1}} \right ) }-{\frac {i/2s\pi }{m-1} \left ( {\it csgn} \left ( {\frac {i}{n-1} \left ( \left ( -{\frac {n \left ( {x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a \right ) }{a \left ( n-1 \right ) }}+{\frac {{x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a}{a \left ( n-1 \right ) }} \right ) a+{x}^{-n+1}nm-{x}^{-n+1}n \right ) } \right ) \right ) ^{2}{\it csgn} \left ( i \left ( \left ( -{\frac {n \left ( {x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a \right ) }{a \left ( n-1 \right ) }}+{\frac {{x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a}{a \left ( n-1 \right ) }} \right ) a+{x}^{-n+1}nm-{x}^{-n+1}n \right ) \right ) }+{\frac {i/2s\pi }{m-1}{\it csgn} \left ( {\frac {i}{n-1} \left ( \left ( -{\frac {n \left ( {x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a \right ) }{a \left ( n-1 \right ) }}+{\frac {{x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a}{a \left ( n-1 \right ) }} \right ) a+{x}^{-n+1}nm-{x}^{-n+1}n \right ) } \right ) {\it csgn} \left ( {\frac {i}{n-1}} \right ) {\it csgn} \left ( i \left ( \left ( -{\frac {n \left ( {x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a \right ) }{a \left ( n-1 \right ) }}+{\frac {{x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a}{a \left ( n-1 \right ) }} \right ) a+{x}^{-n+1}nm-{x}^{-n+1}n \right ) \right ) }-{\frac {i/2s\pi }{m-1}{\it csgn} \left ( {\frac {i}{n-1} \left ( \left ( -{\frac {n \left ( {x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a \right ) }{a \left ( n-1 \right ) }}+{\frac {{x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a}{a \left ( n-1 \right ) }} \right ) a+{x}^{-n+1}nm-{x}^{-n+1}n \right ) } \right ) \left ( {\it csgn} \left ( {\frac {i}{a \left ( n-1 \right ) } \left ( \left ( -{\frac {n \left ( {x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a \right ) }{a \left ( n-1 \right ) }}+{\frac {{x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a}{a \left ( n-1 \right ) }} \right ) a+{x}^{-n+1}nm-{x}^{-n+1}n \right ) } \right ) \right ) ^{2}}+{\frac {i/2s\pi }{m-1}{\it csgn} \left ( {\frac {i}{n-1} \left ( \left ( -{\frac {n \left ( {x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a \right ) }{a \left ( n-1 \right ) }}+{\frac {{x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a}{a \left ( n-1 \right ) }} \right ) a+{x}^{-n+1}nm-{x}^{-n+1}n \right ) } \right ) {\it csgn} \left ( {\frac {i}{a \left ( n-1 \right ) } \left ( \left ( -{\frac {n \left ( {x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a \right ) }{a \left ( n-1 \right ) }}+{\frac {{x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a}{a \left ( n-1 \right ) }} \right ) a+{x}^{-n+1}nm-{x}^{-n+1}n \right ) } \right ) {\it csgn} \left ( {\frac {i}{a}} \right ) }+{\frac {i/2s\pi }{m-1} \left ( {\it csgn} \left ( {\frac {i}{a \left ( n-1 \right ) } \left ( \left ( -{\frac {n \left ( {x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a \right ) }{a \left ( n-1 \right ) }}+{\frac {{x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a}{a \left ( n-1 \right ) }} \right ) a+{x}^{-n+1}nm-{x}^{-n+1}n \right ) } \right ) \right ) ^{3}}-{\frac {i/2s\pi }{m-1} \left ( {\it csgn} \left ( {\frac {i}{a \left ( n-1 \right ) } \left ( \left ( -{\frac {n \left ( {x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a \right ) }{a \left ( n-1 \right ) }}+{\frac {{x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a}{a \left ( n-1 \right ) }} \right ) a+{x}^{-n+1}nm-{x}^{-n+1}n \right ) } \right ) \right ) ^{2}{\it csgn} \left ( {\frac {i}{a}} \right ) }}}}+{\it \_F1} \left ( -{\frac {{x}^{-n+1}nm-{y}^{1-m}an-{x}^{-n+1}n+{y}^{1-m}a}{a \left ( n-1 \right ) }} \right ) \] Result has unresolved integral
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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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h]; 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 \frac {e^{-\frac {b s x^{m-n+1}}{m a-n a+a}} x^{-n} \left (-\frac {b s x^{m-n+1}}{a (m-n+1)}\right )^{-\frac {k}{m-n+1}-\frac {1}{m-n+1}} \left (-c \left (-\frac {b s x^{m-n+1}}{a (m-n+1)}\right )^{\frac {n}{m-n+1}} \left (y^{\frac {m a}{m a-n a+a}-\frac {n a}{m a-n a+a}+\frac {a}{m a-n a+a}}\right )^s \text {Gamma}\left (\frac {k-n+1}{m-n+1},-\frac {b s x^{m-n+1}}{a (m-n+1)}\right ) x^{k+1}+c n \left (-\frac {b s x^{m-n+1}}{a (m-n+1)}\right )^{\frac {n}{m-n+1}} \left (y^{\frac {m a}{m a-n a+a}-\frac {n a}{m a-n a+a}+\frac {a}{m a-n a+a}}\right )^s \text {Gamma}\left (\frac {k-n+1}{m-n+1},-\frac {b s x^{m-n+1}}{a (m-n+1)}\right ) x^{k+1}+a e^{\frac {b s x^{m-n+1}}{m a-n a+a}} \left (-\frac {b s x^{m-n+1}}{a (m-n+1)}\right )^{\frac {k}{m-n+1}+\frac {1}{m-n+1}} c_1\left (e^{\frac {-b x^{m-n+1}+a \log (y)+a m \log (y)-a n \log (y)}{m a-n a+a}}\right ) x^n+a e^{\frac {b s x^{m-n+1}}{m a-n a+a}} n^2 \left (-\frac {b s x^{m-n+1}}{a (m-n+1)}\right )^{\frac {k}{m-n+1}+\frac {1}{m-n+1}} c_1\left (e^{\frac {-b x^{m-n+1}+a \log (y)+a m \log (y)-a n \log (y)}{m a-n a+a}}\right ) x^n+a e^{\frac {b s x^{m-n+1}}{m a-n a+a}} m \left (-\frac {b s x^{m-n+1}}{a (m-n+1)}\right )^{\frac {k}{m-n+1}+\frac {1}{m-n+1}} c_1\left (e^{\frac {-b x^{m-n+1}+a \log (y)+a m \log (y)-a n \log (y)}{m a-n a+a}}\right ) x^n-2 a e^{\frac {b s x^{m-n+1}}{m a-n a+a}} n \left (-\frac {b s x^{m-n+1}}{a (m-n+1)}\right )^{\frac {k}{m-n+1}+\frac {1}{m-n+1}} c_1\left (e^{\frac {-b x^{m-n+1}+a \log (y)+a m \log (y)-a n \log (y)}{m a-n a+a}}\right ) x^n-a e^{\frac {b s x^{m-n+1}}{m a-n a+a}} m n \left (-\frac {b s x^{m-n+1}}{a (m-n+1)}\right )^{\frac {k}{m-n+1}+\frac {1}{m-n+1}} c_1\left (e^{\frac {-b x^{m-n+1}+a \log (y)+a m \log (y)-a n \log (y)}{m a-n a+a}}\right ) x^n+d e^{\frac {b s x^{m-n+1}}{m a-n a+a}} \left (-\frac {b s x^{m-n+1}}{a (m-n+1)}\right )^{\frac {k}{m-n+1}+\frac {1}{m-n+1}} x+d e^{\frac {b s x^{m-n+1}}{m a-n a+a}} m \left (-\frac {b s x^{m-n+1}}{a (m-n+1)}\right )^{\frac {k}{m-n+1}+\frac {1}{m-n+1}} x-d e^{\frac {b s x^{m-n+1}}{m a-n a+a}} n \left (-\frac {b s x^{m-n+1}}{a (m-n+1)}\right )^{\frac {k}{m-n+1}+\frac {1}{m-n+1}} x\right )}{a (n-1) (-m+n-1)}\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 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'));
\[ w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{a} \left ( {{\it \_a}}^{k-n}c \left ( y{{\rm e}^{-{\frac {{x}^{-n+m+1}n}{a \left ( -n+m+1 \right ) }}+{\frac {{{\it \_a}}^{-n+m+1}n}{a \left ( -n+m+1 \right ) }}}} \right ) ^{s}+{{\it \_a}}^{-n}d \right ) }{d{\it \_a}}+{\it \_F1} \left ( y{{\rm e}^{-{\frac {{x}^{-n+m+1}n}{a \left ( -n+m+1 \right ) }}}} \right ) \] Result has unresolved integral
____________________________________________________________________________________
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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p]; 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]];
\[ \text {\$Aborted} \] Timed out
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p'; 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'));
\[ w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{a} \left ( {{\it \_a}}^{-n+p}s \left ( y{{\rm e}^{-{\frac {{x}^{-n+m+1}n}{a \left ( -n+m+1 \right ) }}+{\frac {{{\it \_a}}^{-n+m+1}n}{a \left ( -n+m+1 \right ) }}}} \right ) ^{q}+{{\it \_a}}^{-n}d \right ) }{d{\it \_a}}+{\it \_F1} \left ( y{{\rm e}^{-{\frac {{x}^{-n+m+1}n}{a \left ( -n+m+1 \right ) }}}} \right ) \]
____________________________________________________________________________________
Added Feb. 9, 2019.
Problem Chapter 3.2.4.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x^n w_x +(b x^m y^k + c x^l y) w_y = s x^p y^q + d \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p]; pde = a*x^n*D[w[x, y], x] + (b*x^m*y^k + c*x^l*y)*D[w[x, y], y] == s*x^p*y^q + d; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {\$Aborted} \] Timed out
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; pde :=a*x^n*diff(w(x,y),x) +(b*x^m*y^k + c*x^l*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 {1}{a} \left ( {{\it \_a}}^{-n+p}s \left ( \left ( -{\frac {1}{a} \left ( {\frac {kb}{-n+l+1} \left ( {\frac {c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{-{\frac {-n+m+1}{-n+l+1}}} \left ( {\frac { \left ( -n+l+1 \right ) ^{2}{{\it \_a}}^{m-l}a}{ \left ( -n+m+1 \right ) \left ( l+m-2\,n+2 \right ) \left ( 2\,l+m-3\,n+3 \right ) c \left ( 1-k \right ) } \left ( {\frac {c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{{\frac {-n+m+1}{-n+l+1}}} \left ( {\frac {{{\it \_a}}^{-n+l+1}c \left ( 1-k \right ) {l}^{2}}{ \left ( -n+l+1 \right ) a}}-2\,{\frac {{{\it \_a}}^{-n+l+1}c \left ( 1-k \right ) ln}{ \left ( -n+l+1 \right ) a}}+{\frac {{{\it \_a}}^{-n+l+1}c \left ( 1-k \right ) {n}^{2}}{ \left ( -n+l+1 \right ) a}}+2\,{\frac {{{\it \_a}}^{-n+l+1}c \left ( 1-k \right ) l}{ \left ( -n+l+1 \right ) a}}-2\,{\frac {{{\it \_a}}^{-n+l+1}c \left ( 1-k \right ) n}{ \left ( -n+l+1 \right ) a}}+{l}^{2}+ml-3\,nl-mn+2\,{n}^{2}+{\frac {{{\it \_a}}^{-n+l+1}c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}}+3\,l+m-4\,n+2 \right ) \left ( {\frac {{{\it \_a}}^{-n+l+1}c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{-1/2\,{\frac {l+m-2\,n+2}{-n+l+1}}}{{\rm e}^{-1/2\,{\frac {{{\it \_a}}^{-n+l+1}c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}}}} \WhittakerM \left ( {\frac {-n+m+1}{-n+l+1}}-1/2\,{\frac {l+m-2\,n+2}{-n+l+1}},1/2\,{\frac {l+m-2\,n+2}{-n+l+1}}+1/2,{\frac {{{\it \_a}}^{-n+l+1}c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) }+{\frac { \left ( -n+l+1 \right ) ^{2}{{\it \_a}}^{m-l} \left ( l+m-2\,n+2 \right ) a}{ \left ( -n+m+1 \right ) \left ( 2\,l+m-3\,n+3 \right ) c \left ( 1-k \right ) } \left ( {\frac {c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{{\frac {-n+m+1}{-n+l+1}}} \left ( {\frac {{{\it \_a}}^{-n+l+1}c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{-1/2\,{\frac {l+m-2\,n+2}{-n+l+1}}}{{\rm e}^{-1/2\,{\frac {{{\it \_a}}^{-n+l+1}c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}}}} \WhittakerM \left ( {\frac {-n+m+1}{-n+l+1}}-1/2\,{\frac {l+m-2\,n+2}{-n+l+1}}+1,1/2\,{\frac {l+m-2\,n+2}{-n+l+1}}+1/2,{\frac {{{\it \_a}}^{-n+l+1}c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) } \right ) }-{\frac {b}{-n+l+1} \left ( {\frac {c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{-{\frac {-n+m+1}{-n+l+1}}} \left ( {\frac { \left ( -n+l+1 \right ) ^{2}{{\it \_a}}^{m-l}a}{ \left ( -n+m+1 \right ) \left ( l+m-2\,n+2 \right ) \left ( 2\,l+m-3\,n+3 \right ) c \left ( 1-k \right ) } \left ( {\frac {c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{{\frac {-n+m+1}{-n+l+1}}} \left ( {\frac {{{\it \_a}}^{-n+l+1}c \left ( 1-k \right ) {l}^{2}}{ \left ( -n+l+1 \right ) a}}-2\,{\frac {{{\it \_a}}^{-n+l+1}c \left ( 1-k \right ) ln}{ \left ( -n+l+1 \right ) a}}+{\frac {{{\it \_a}}^{-n+l+1}c \left ( 1-k \right ) {n}^{2}}{ \left ( -n+l+1 \right ) a}}+2\,{\frac {{{\it \_a}}^{-n+l+1}c \left ( 1-k \right ) l}{ \left ( -n+l+1 \right ) a}}-2\,{\frac {{{\it \_a}}^{-n+l+1}c \left ( 1-k \right ) n}{ \left ( -n+l+1 \right ) a}}+{l}^{2}+ml-3\,nl-mn+2\,{n}^{2}+{\frac {{{\it \_a}}^{-n+l+1}c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}}+3\,l+m-4\,n+2 \right ) \left ( {\frac {{{\it \_a}}^{-n+l+1}c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{-1/2\,{\frac {l+m-2\,n+2}{-n+l+1}}}{{\rm e}^{-1/2\,{\frac {{{\it \_a}}^{-n+l+1}c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}}}} \WhittakerM \left ( {\frac {-n+m+1}{-n+l+1}}-1/2\,{\frac {l+m-2\,n+2}{-n+l+1}},1/2\,{\frac {l+m-2\,n+2}{-n+l+1}}+1/2,{\frac {{{\it \_a}}^{-n+l+1}c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) }+{\frac { \left ( -n+l+1 \right ) ^{2}{{\it \_a}}^{m-l} \left ( l+m-2\,n+2 \right ) a}{ \left ( -n+m+1 \right ) \left ( 2\,l+m-3\,n+3 \right ) c \left ( 1-k \right ) } \left ( {\frac {c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{{\frac {-n+m+1}{-n+l+1}}} \left ( {\frac {{{\it \_a}}^{-n+l+1}c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{-1/2\,{\frac {l+m-2\,n+2}{-n+l+1}}}{{\rm e}^{-1/2\,{\frac {{{\it \_a}}^{-n+l+1}c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}}}} \WhittakerM \left ( {\frac {-n+m+1}{-n+l+1}}-1/2\,{\frac {l+m-2\,n+2}{-n+l+1}}+1,1/2\,{\frac {l+m-2\,n+2}{-n+l+1}}+1/2,{\frac {{{\it \_a}}^{-n+l+1}c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) } \right ) }-kb\int \!{{\rm e}^{{\frac {{x}^{-n+l+1}c \left ( k-1 \right ) }{ \left ( -n+l+1 \right ) a}}}}{x}^{-n+m}\,{\rm d}x-{{y}^{ \left ( -n+l+1 \right ) ^{-1}}a{{\rm e}^{{\frac {c{x}^{-n+l+1}k}{ \left ( -n+l+1 \right ) a}}}}{y}^{{\frac {kn}{-n+l+1}}}{y}^{{\frac {l}{-n+l+1}}} \left ( {y}^{{\frac {kl}{-n+l+1}}} \right ) ^{-1} \left ( {y}^{{\frac {k}{-n+l+1}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {c{x}^{-n+l+1}}{ \left ( -n+l+1 \right ) a}}}} \right ) ^{-1} \left ( {y}^{{\frac {n}{-n+l+1}}} \right ) ^{-1}}+b\int \!{{\rm e}^{{\frac {{x}^{-n+l+1}c \left ( k-1 \right ) }{ \left ( -n+l+1 \right ) a}}}}{x}^{-n+m}\,{\rm d}x \right ) } \right ) ^{- \left ( k-1 \right ) ^{-1}}{{\rm e}^{{\frac {{{\it \_a}}^{-n+l+1}c}{ \left ( -n+l+1 \right ) a}}}} \right ) ^{q}+{{\it \_a}}^{-n}d \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {1}{a} \left ( {\frac {kb}{-n+l+1} \left ( {\frac {c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{-{\frac {-n+m+1}{-n+l+1}}} \left ( {\frac { \left ( -n+l+1 \right ) ^{2}{x}^{m-l}a}{ \left ( -n+m+1 \right ) \left ( l+m-2\,n+2 \right ) \left ( 2\,l+m-3\,n+3 \right ) c \left ( 1-k \right ) } \left ( {\frac {c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{{\frac {-n+m+1}{-n+l+1}}} \left ( {\frac {c{x}^{-n+l+1} \left ( 1-k \right ) {l}^{2}}{ \left ( -n+l+1 \right ) a}}-2\,{\frac {c{x}^{-n+l+1} \left ( 1-k \right ) ln}{ \left ( -n+l+1 \right ) a}}+{\frac {c{x}^{-n+l+1} \left ( 1-k \right ) {n}^{2}}{ \left ( -n+l+1 \right ) a}}+2\,{\frac {c{x}^{-n+l+1} \left ( 1-k \right ) l}{ \left ( -n+l+1 \right ) a}}-2\,{\frac {c{x}^{-n+l+1} \left ( 1-k \right ) n}{ \left ( -n+l+1 \right ) a}}+{l}^{2}+ml-3\,nl-mn+2\,{n}^{2}+{\frac {c{x}^{-n+l+1} \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}}+3\,l+m-4\,n+2 \right ) \left ( {\frac {c{x}^{-n+l+1} \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{-1/2\,{\frac {l+m-2\,n+2}{-n+l+1}}}{{\rm e}^{-1/2\,{\frac {c{x}^{-n+l+1} \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}}}} \WhittakerM \left ( {\frac {-n+m+1}{-n+l+1}}-1/2\,{\frac {l+m-2\,n+2}{-n+l+1}},1/2\,{\frac {l+m-2\,n+2}{-n+l+1}}+1/2,{\frac {c{x}^{-n+l+1} \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) }+{\frac { \left ( -n+l+1 \right ) ^{2}{x}^{m-l} \left ( l+m-2\,n+2 \right ) a}{ \left ( -n+m+1 \right ) \left ( 2\,l+m-3\,n+3 \right ) c \left ( 1-k \right ) } \left ( {\frac {c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{{\frac {-n+m+1}{-n+l+1}}} \left ( {\frac {c{x}^{-n+l+1} \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{-1/2\,{\frac {l+m-2\,n+2}{-n+l+1}}}{{\rm e}^{-1/2\,{\frac {c{x}^{-n+l+1} \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}}}} \WhittakerM \left ( {\frac {-n+m+1}{-n+l+1}}-1/2\,{\frac {l+m-2\,n+2}{-n+l+1}}+1,1/2\,{\frac {l+m-2\,n+2}{-n+l+1}}+1/2,{\frac {c{x}^{-n+l+1} \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) } \right ) }+{{y}^{ \left ( -n+l+1 \right ) ^{-1}}a{{\rm e}^{{\frac {c{x}^{-n+l+1}k}{ \left ( -n+l+1 \right ) a}}}}{y}^{{\frac {kn}{-n+l+1}}}{y}^{{\frac {l}{-n+l+1}}} \left ( {y}^{{\frac {kl}{-n+l+1}}} \right ) ^{-1} \left ( {y}^{{\frac {k}{-n+l+1}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {c{x}^{-n+l+1}}{ \left ( -n+l+1 \right ) a}}}} \right ) ^{-1} \left ( {y}^{{\frac {n}{-n+l+1}}} \right ) ^{-1}}-{\frac {b}{-n+l+1} \left ( {\frac {c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{-{\frac {-n+m+1}{-n+l+1}}} \left ( {\frac { \left ( -n+l+1 \right ) ^{2}{x}^{m-l}a}{ \left ( -n+m+1 \right ) \left ( l+m-2\,n+2 \right ) \left ( 2\,l+m-3\,n+3 \right ) c \left ( 1-k \right ) } \left ( {\frac {c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{{\frac {-n+m+1}{-n+l+1}}} \left ( {\frac {c{x}^{-n+l+1} \left ( 1-k \right ) {l}^{2}}{ \left ( -n+l+1 \right ) a}}-2\,{\frac {c{x}^{-n+l+1} \left ( 1-k \right ) ln}{ \left ( -n+l+1 \right ) a}}+{\frac {c{x}^{-n+l+1} \left ( 1-k \right ) {n}^{2}}{ \left ( -n+l+1 \right ) a}}+2\,{\frac {c{x}^{-n+l+1} \left ( 1-k \right ) l}{ \left ( -n+l+1 \right ) a}}-2\,{\frac {c{x}^{-n+l+1} \left ( 1-k \right ) n}{ \left ( -n+l+1 \right ) a}}+{l}^{2}+ml-3\,nl-mn+2\,{n}^{2}+{\frac {c{x}^{-n+l+1} \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}}+3\,l+m-4\,n+2 \right ) \left ( {\frac {c{x}^{-n+l+1} \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{-1/2\,{\frac {l+m-2\,n+2}{-n+l+1}}}{{\rm e}^{-1/2\,{\frac {c{x}^{-n+l+1} \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}}}} \WhittakerM \left ( {\frac {-n+m+1}{-n+l+1}}-1/2\,{\frac {l+m-2\,n+2}{-n+l+1}},1/2\,{\frac {l+m-2\,n+2}{-n+l+1}}+1/2,{\frac {c{x}^{-n+l+1} \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) }+{\frac { \left ( -n+l+1 \right ) ^{2}{x}^{m-l} \left ( l+m-2\,n+2 \right ) a}{ \left ( -n+m+1 \right ) \left ( 2\,l+m-3\,n+3 \right ) c \left ( 1-k \right ) } \left ( {\frac {c \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{{\frac {-n+m+1}{-n+l+1}}} \left ( {\frac {c{x}^{-n+l+1} \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) ^{-1/2\,{\frac {l+m-2\,n+2}{-n+l+1}}}{{\rm e}^{-1/2\,{\frac {c{x}^{-n+l+1} \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}}}} \WhittakerM \left ( {\frac {-n+m+1}{-n+l+1}}-1/2\,{\frac {l+m-2\,n+2}{-n+l+1}}+1,1/2\,{\frac {l+m-2\,n+2}{-n+l+1}}+1/2,{\frac {c{x}^{-n+l+1} \left ( 1-k \right ) }{ \left ( -n+l+1 \right ) a}} \right ) } \right ) } \right ) } \right ) \]
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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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p]; 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 (\frac {a (m+1)}{(k+1) \left (b x^{m+1}+\frac {a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{k+1}\right )}\right )^{-\frac {1}{k+1}}\right )^{-k} \left (a b c_1\left (\frac {a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{a (k+1) (m+1)}\right ) \left (\left (\frac {a (m+1)}{(k+1) \left (b x^{m+1}+\frac {a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{k+1}\right )}\right )^{-\frac {1}{k+1}}\right )^k+a b m c_1\left (\frac {a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{a (k+1) (m+1)}\right ) \left (\left (\frac {a (m+1)}{(k+1) \left (b x^{m+1}+\frac {a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{k+1}\right )}\right )^{-\frac {1}{k+1}}\right )^k-a c y^{k+1} \left (\frac {b x^{m+1}}{\frac {m \left (a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}\right )}{(k+1) (m+1)}+\frac {a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{(k+1) (m+1)}}+1\right )^{\frac {k}{k+1}}-a c m y^{k+1} \left (\frac {b x^{m+1}}{\frac {m \left (a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}\right )}{(k+1) (m+1)}+\frac {a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{(k+1) (m+1)}}+1\right )^{\frac {k}{k+1}}+b c x^{m+1} \left (\frac {b x^{m+1}}{\frac {m \left (a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}\right )}{(k+1) (m+1)}+\frac {a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{(k+1) (m+1)}}+1\right )^{\frac {k}{k+1}}+b c k x^{m+1} \left (\frac {b x^{m+1}}{\frac {m \left (a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}\right )}{(k+1) (m+1)}+\frac {a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{(k+1) (m+1)}}+1\right )^{\frac {k}{k+1}}+a c y^{k+1}+a c m y^{k+1}+b d x \left (\frac {b x^{m+1}}{\frac {m \left (a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}\right )}{(k+1) (m+1)}+\frac {a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{(k+1) (m+1)}}+1\right )^{\frac {k}{k+1}} \text {Hypergeometric2F1}\left (\frac {k}{k+1},\frac {1}{m+1},1+\frac {1}{m+1},-\frac {b x^{m+1}}{\frac {m \left (a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}\right )}{(k+1) (m+1)}+\frac {a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{(k+1) (m+1)}}\right )+b d m x \left (\frac {b x^{m+1}}{\frac {m \left (a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}\right )}{(k+1) (m+1)}+\frac {a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{(k+1) (m+1)}}+1\right )^{\frac {k}{k+1}} \text {Hypergeometric2F1}\left (\frac {k}{k+1},\frac {1}{m+1},1+\frac {1}{m+1},-\frac {b x^{m+1}}{\frac {m \left (a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}\right )}{(k+1) (m+1)}+\frac {a y^{k+1}+a m y^{k+1}-b x^{m+1}-b k x^{m+1}}{(k+1) (m+1)}}\right )\right )}{a b (m+1)}\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 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'));
\[ w \left ( x,y \right ) =\int ^{x}\!{\frac {c{{\it \_a}}^{m}+d}{a} \left ( \left ( {\frac {1}{a \left ( n+1 \right ) } \left ( {{\it \_a}}^{n+1}bk+{\frac { \left ( -{x}^{n+1}bk+{y}^{k+1}an-{x}^{n+1}b+{y}^{k+1}a \right ) n}{n+1}}+{{\it \_a}}^{n+1}b+{\frac {-{x}^{n+1}bk+{y}^{k+1}an-{x}^{n+1}b+{y}^{k+1}a}{n+1}} \right ) } \right ) ^{ \left ( k+1 \right ) ^{-1}} \right ) ^{-k}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {-{x}^{n+1}bk+{y}^{k+1}an-{x}^{n+1}b+{y}^{k+1}a}{a \left ( n+1 \right ) }} \right ) \]