____________________________________________________________________________________
Added Feb. 17, 2019.
Problem Chapter 4.2.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = (c x^n + d y^m) w \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == (c*x^n + d*y^m)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {a y-b x}{a}\right ) \exp \left (\frac {d \left (\frac {a y-b x}{b (m+1)}+\frac {b x}{b m+b}\right ) \left (\frac {a y-b x}{a}+\frac {b x}{a}\right )^m}{a}+\frac {c x^{n+1}}{a (n+1)}\right )\right \}\right \} \]
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'; g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; pde :=a*diff(w(x,y),x)+b*diff(w(x,y),y) = (c*x^n + d*y^m)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ \text {Bad latex generated} \]
____________________________________________________________________________________
Added Feb. 17, 2019.
Problem Chapter 4.2.4.2 case \(n eq -1, n eq -2\), from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c x^n y w \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*x^n*y*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; sol[[2]] = Assuming[{n != -1, n != -2}, Simplify[sol[[2]]]];
\[ \left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {c x^{n+1} (a (n+2) y-b x)}{a^2 (n+1) (n+2)}\right )\right \}\right \} \]
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'; g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; pde :=a*diff(w(x,y),x)+b*diff(w(x,y),y) = c*x^n*y*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); sol:=simplify(sol) assuming n<>-1, n<>-2
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {{\rm e}^{{\frac { \left ( ay \left ( n+2 \right ) {x}^{n+1}-{x}^{n+2}b \right ) c}{ \left ( n+2 \right ) \left ( n+1 \right ) {a}^{2}}}}} \]
____________________________________________________________________________________
Added Feb. 17, 2019.
Problem Chapter 4.2.4.2 case \(n= -1\), from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c x^n y w \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*x^n*y*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}, Assumptions -> n == -1], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {a y-b x}{a}\right ) e^{\frac {c (\log (x) (a y-b x)+b x)}{a^2}}\right \}\right \} \]
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';q:='q';p:='p';l:='l'; g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; pde :=a*diff(w(x,y),x)+b*diff(w(x,y),y) = c*x^n*y*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) assuming n=-1),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {x}^{{\frac { \left ( ya-bx \right ) c}{{a}^{2}}}}{{\rm e}^{{\frac {bcx}{{a}^{2}}}}} \]
____________________________________________________________________________________
Added Feb. 17, 2019.
Problem Chapter 4.2.4.2 case \(n= -2\), from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c x^n y w \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*x^n*y*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}, Assumptions -> n == -2], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {a y-b x}{a}\right ) e^{\frac {c \left (b \log (x)-\frac {a y-b x}{x}\right )}{a^2}}\right \}\right \} \]
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';q:='q';p:='p';l:='l'; g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; pde :=a*diff(w(x,y),x)+b*diff(w(x,y),y) = c*x^n*y*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) assuming n=-2),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {{\rm e}^{-{\frac { \left ( ya-bx \right ) c}{{a}^{2}x}}}}{x}^{{\frac {bc}{{a}^{2}}}} \]
____________________________________________________________________________________
Added Feb. 17, 2019.
Problem Chapter 4.2.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + y w_y = a(x^2+y^2)^k w \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; pde = x*D[w[x, y], x] + y*D[w[x, y], y] == a*(x^2 + y^2)^k*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {y}{x}\right ) e^{\frac {a \left (x^2 \left (\frac {y^2}{x^2}+1\right )\right )^k}{2 k}}\right \}\right \} \]
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';q:='q';p:='p';l:='l'; g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; pde :=x*diff(w(x,y),x)+y*diff(w(x,y),y) = a*(x^2+y^2)^k*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {y}{x}} \right ) {{\rm e}^{1/2\,{\frac {a}{k} \left ( {x}^{2} \left ( {\frac {{y}^{2}}{{x}^{2}}}+1 \right ) \right ) ^{k}}}} \]
____________________________________________________________________________________
Added Feb. 17, 2019.
Problem Chapter 4.2.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x w_x + b y w_y = c x^n y^m w \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == c*x^n*y^m*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (y x^{-\frac {b}{a}}\right ) e^{\frac {c y^m x^n}{a \left (\frac {b m}{a}+n\right )}}\right \}\right \} \]
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'; g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; pde :=a*x*diff(w(x,y),x)+b*y*diff(w(x,y),y) = c*x^n*y^m*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) {{\rm e}^{{\frac {c{x}^{n}{y}^{m}}{an+bm}}}} \]
____________________________________________________________________________________
Added Feb. 17, 2019.
Problem Chapter 4.2.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x w_x + b y w_y = (c x^n + k y^m) w \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == (c*x^n + k*y^m)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (y x^{-\frac {b}{a}}\right ) e^{\frac {c x^n}{a n}+\frac {k y^m}{b m}}\right \}\right \} \]
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';q:='q';p:='p';l:='l'; g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; pde :=a*x*diff(w(x,y),x)+b*y*diff(w(x,y),y) = (c*x^n + k*y^m)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) {{\rm e}^{{\frac {k{y}^{m}an+{x}^{n}cbm}{abmn}}}} \]
____________________________________________________________________________________
Added Feb. 17, 2019.
Problem Chapter 4.2.4.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ m x w_x + n y w_y = (a x^n + b y^m)^k w \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; pde = m*x*D[w[x, y], x] + n*y*D[w[x, y], y] == (a*x^n + b*y^m)^k*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (y x^{-\frac {n}{m}}\right ) e^{\frac {\left (a x^n+b y^m\right )^k}{k m n}}\right \}\right \} \]
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';q:='q';p:='p';l:='l'; g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; pde :=m*x*diff(w(x,y),x)+n*y*diff(w(x,y),y) = (a*x^n + b*y^m)^k*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( y{x}^{-{\frac {n}{m}}} \right ) {{\rm e}^{{\frac { \left ( {x}^{n}a+{y}^{m}b \right ) ^{k}}{knm}}}} \]
____________________________________________________________________________________
Added Feb. 17, 2019.
Problem Chapter 4.2.4.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x^n w_x + b y^m w_y = (c x^k + d y^s) w \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; pde = a*x^n*D[w[x, y], x] + b*y^m*D[w[x, y], y] == (c*x^k + d*y^s)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; sol[[2]] = Simplify[sol[[2]]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {b x^{1-n}}{a (n-1)}-\frac {y^{1-m}}{m-1}\right ) \exp \left (\frac {c x^{k-n+1}}{a k-a n+a}-\frac {d y^{1-m} \left (\left (y^{m-1}\right )^{\frac {1}{m-1}}\right )^s}{b (m-s-1)}\right )\right \}\right \} \]
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'; g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; pde :=a*x^n*diff(w(x,y),x)+b*y^m*diff(w(x,y),y) = (c*x^k + d*y^s)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); sol:=simplify(sol);
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-{x}^{-n+1}b \left ( m-1 \right ) +a \left ( n-1 \right ) {y}^{1-m}}{a \left ( n-1 \right ) }} \right ) {{\rm e}^{{\frac {1}{b \left ( m-s-1 \right ) a \left ( k-n+1 \right ) } \left ( -{y}^{1-m} \left ( n-1 \right ) ^{{\frac {s}{m-1}}}{a}^{{\frac {m+s-1}{m-1}}} \left ( a \left ( n-1 \right ) {y}^{1-m} \right ) ^{-{\frac {s}{m-1}}}d \left ( k-n+1 \right ) {{\rm e}^{{\frac {i/2\pi \,s}{m-1} \left ( -{\it csgn} \left ( i \left ( n-1 \right ) {y}^{1-m}a \right ) -{\it csgn} \left ( {\frac {i}{n-1}} \right ) +{\it csgn} \left ( i{y}^{1-m}a \right ) {\it csgn} \left ( i \left ( n-1 \right ) {y}^{1-m}a \right ) {\it csgn} \left ( {\frac {i}{n-1}} \right ) +{\it csgn} \left ( i{y}^{1-m}a \right ) {\it csgn} \left ( i{y}^{1-m} \right ) {\it csgn} \left ( {\frac {i}{a}} \right ) +{\it csgn} \left ( i{y}^{1-m} \right ) -{\it csgn} \left ( {\frac {i}{a}} \right ) \right ) }}}+c{x}^{k-n+1}b \left ( m-s-1 \right ) \right ) }}} \]
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Added Feb. 17, 2019.
Problem Chapter 4.2.4.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x^n w_x + b x^m y w_y = (c x^k y^s + d) w \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; pde = a*x^n*D[w[x, y], x] + b*x^m*y*D[w[x, y], y] == (c*x^k*y^s + d)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; sol[[2]] = Simplify[sol[[2]]];
\[ \left \{\left \{w(x,y)\to c_1\left (y e^{-\frac {b x^{m-n+1}}{a m-a n+a}}\right ) \exp \left (\frac {x^{1-n} \left (\frac {d}{1-n}-\frac {c x^k y^s e^{-\frac {b s x^{m-n+1}}{a m-a n+a}} \left (-\frac {b s x^{m-n+1}}{a m-a n+a}\right )^{\frac {-k+n-1}{m-n+1}} \text {Gamma}\left (\frac {k-n+1}{m-n+1},-\frac {b s x^{m-n+1}}{a m-a n+a}\right )}{m-n+1}\right )}{a}\right )\right \}\right \} \]
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'; g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; pde :=a*x^n*diff(w(x,y),x)+b*x^m*y*diff(w(x,y),y) = (c*x^k*y^s + d)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); sol:=simplify(sol);
\[ w \left ( x,y \right ) ={\it \_F1} \left ( y{{\rm e}^{-{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{a} \left ( {{\it \_a}}^{k-n}c \left ( y{{\rm e}^{-{\frac {b \left ( {x}^{-n+m+1}-{{\it \_a}}^{-n+m+1} \right ) }{a \left ( -n+m+1 \right ) }}}} \right ) ^{s}+{{\it \_a}}^{-n}d \right ) }{d{\it \_a}}}} \]
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Added Feb. 17, 2019.
Problem Chapter 4.2.4.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x^n w_x + (b x^m y+c x^k) w_y = (s x^p y^q + d) w \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; 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)*w[x, y]; 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'; g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; 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 + d)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); sol:=simplify(sol);
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{ab \left ( k+2\,m-3\,n+3 \right ) \left ( k+m-2\,n+2 \right ) \left ( k-n+1 \right ) } \left ( -a{{\rm e}^{-1/2\,{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \left ( {\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) ^{{\frac {-k-m+2\,n-2}{-2\,n+2\,m+2}}}c{x}^{k-m} \left ( -n+m+1 \right ) \left ( k+m-2\,n+2 \right ) ^{2} \WhittakerM \left ( {\frac {k+m-2\,n+2}{-2\,n+2\,m+2}},{\frac {k+2\,m-3\,n+3}{-2\,n+2\,m+2}},{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) - \left ( {\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) ^{{\frac {-k-m+2\,n-2}{-2\,n+2\,m+2}}} \left ( {x}^{k-n+1}b+{x}^{k-m}a \left ( k+m-2\,n+2 \right ) \right ) \left ( -n+m+1 \right ) ^{2}{{\rm e}^{-1/2\,{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}}c \WhittakerM \left ( {\frac {k-m}{-2\,n+2\,m+2}},{\frac {k+2\,m-3\,n+3}{-2\,n+2\,m+2}},{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) +{{\rm e}^{-{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}}ayb \left ( k-n+1 \right ) \left ( k+2\,m-3\,n+3 \right ) \left ( k+m-2\,n+2 \right ) \right ) } \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{a} \left ( \left ( {\frac {1}{ab \left ( k+2\,m-3\,n+3 \right ) \left ( k+m-2\,n+2 \right ) \left ( k-n+1 \right ) } \left ( a{{\it \_a}}^{k-m}{{\rm e}^{1/2\,{\frac {{{\it \_a}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \left ( {\frac {{{\it \_a}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) ^{{\frac {-k-m+2\,n-2}{-2\,n+2\,m+2}}}c \left ( -n+m+1 \right ) \left ( k+m-2\,n+2 \right ) ^{2} \WhittakerM \left ( {\frac {k+m-2\,n+2}{-2\,n+2\,m+2}},{\frac {k+2\,m-3\,n+3}{-2\,n+2\,m+2}},{\frac {{{\it \_a}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) -a{x}^{k-m}{{\rm e}^{-1/2\,{\frac {b \left ( -2\,{{\it \_a}}^{-n+m+1}+{x}^{-n+m+1} \right ) }{a \left ( -n+m+1 \right ) }}}} \left ( {\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) ^{{\frac {-k-m+2\,n-2}{-2\,n+2\,m+2}}}c \left ( -n+m+1 \right ) \left ( k+m-2\,n+2 \right ) ^{2} \WhittakerM \left ( {\frac {k+m-2\,n+2}{-2\,n+2\,m+2}},{\frac {k+2\,m-3\,n+3}{-2\,n+2\,m+2}},{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) + \left ( b{{\it \_a}}^{k-n+1}+{{\it \_a}}^{k-m}a \left ( k+m-2\,n+2 \right ) \right ) \left ( -n+m+1 \right ) ^{2} \left ( {\frac {{{\it \_a}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) ^{{\frac {-k-m+2\,n-2}{-2\,n+2\,m+2}}}{{\rm e}^{1/2\,{\frac {{{\it \_a}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}}c \WhittakerM \left ( {\frac {k-m}{-2\,n+2\,m+2}},{\frac {k+2\,m-3\,n+3}{-2\,n+2\,m+2}},{\frac {{{\it \_a}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) - \left ( -n+m+1 \right ) ^{2}c{{\rm e}^{-1/2\,{\frac {b \left ( -2\,{{\it \_a}}^{-n+m+1}+{x}^{-n+m+1} \right ) }{a \left ( -n+m+1 \right ) }}}} \left ( {\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) ^{{\frac {-k-m+2\,n-2}{-2\,n+2\,m+2}}} \left ( {x}^{k-n+1}b+{x}^{k-m}a \left ( k+m-2\,n+2 \right ) \right ) \WhittakerM \left ( {\frac {k-m}{-2\,n+2\,m+2}},{\frac {k+2\,m-3\,n+3}{-2\,n+2\,m+2}},{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }} \right ) +{{\rm e}^{-{\frac {b \left ( {x}^{-n+m+1}-{{\it \_a}}^{-n+m+1} \right ) }{a \left ( -n+m+1 \right ) }}}}yab \left ( k-n+1 \right ) \left ( k+2\,m-3\,n+3 \right ) \left ( k+m-2\,n+2 \right ) \right ) } \right ) ^{q}s{{\it \_a}}^{-n+p}+{{\it \_a}}^{-n}d \right ) }{d{\it \_a}}}} \]
____________________________________________________________________________________
Added Feb. 17, 2019.
Problem Chapter 4.2.4.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x^n w_x + b x^m y^k w_y = (c x^p y^q + s) w \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; pde = a*x^n*D[w[x, y], x] + b*x^m*y^k*D[w[x, y], y] == (c*x^p*y^q + s)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; sol[[2]] = Simplify[sol[[2]]];
\[ \text {\$Aborted} \] 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'; g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; pde :=a*x^n*diff(w(x,y),x)+b*x^m*y^k*diff(w(x,y),y) = (c*x^p*y^q + s)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); sol:=simplify(sol);
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {b \left ( k-1 \right ) {x}^{-n+m+1}+{y}^{1-k}a \left ( -n+m+1 \right ) }{a \left ( -n+m+1 \right ) }} \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{a} \left ( {{\it \_a}}^{-n+p}c \left ( \left ( {\frac {-b \left ( k-1 \right ) {{\it \_a}}^{-n+m+1}+b \left ( k-1 \right ) {x}^{-n+m+1}+{y}^{1-k}a \left ( -n+m+1 \right ) }{a \left ( -n+m+1 \right ) }} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) ^{q}+{{\it \_a}}^{-n}s \right ) }{d{\it \_a}}}} \]
____________________________________________________________________________________
Added Feb. 17, 2019.
Problem Chapter 4.2.4.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a y^k w_x + b x^n w_y = (c x^m + s) w \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; pde = a*y^k*D[w[x, y], x] + b*x^n*D[w[x, y], y] == (c*x^m + s)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; sol[[2]] = Simplify[sol[[2]]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {y^{k+1}}{k+1}-\frac {b x^{n+1}}{a n+a}\right ) \exp \left (\frac {x \left (\left (y^{-k-1}\right )^{-\frac {1}{k+1}}\right )^{-k} \left (\frac {a (n+1) y^{k+1}}{a (n+1) y^{k+1}-b (k+1) x^{n+1}}\right )^{\frac {k}{k+1}} \left (c x^m \text {Hypergeometric2F1}\left (\frac {k}{k+1},\frac {m+1}{n+1},\frac {m+n+2}{n+1},\frac {b (k+1) x^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )+(m+1) s \text {Hypergeometric2F1}\left (\frac {k}{k+1},\frac {1}{n+1},\frac {1}{n+1}+1,\frac {b (k+1) x^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )\right )}{a (m+1)}\right )\right \}\right \} \]
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'; g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; pde :=a*y^k*diff(w(x,y),x)+b*x^n*diff(w(x,y),y) = (c*x^m+ s)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); sol:=simplify(sol);
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-{x}^{n+1}b \left ( k+1 \right ) +{y}^{k+1}a \left ( n+1 \right ) }{a \left ( n+1 \right ) }} \right ) {{\rm e}^{\int ^{x}\!{\frac {c{{\it \_a}}^{m}+s}{a} \left ( \left ( {\frac {b \left ( k+1 \right ) {{\it \_a}}^{n+1}-{x}^{n+1}b \left ( k+1 \right ) +{y}^{k+1}a \left ( n+1 \right ) }{a \left ( n+1 \right ) }} \right ) ^{ \left ( k+1 \right ) ^{-1}} \right ) ^{-k}}{d{\it \_a}}}} \]
____________________________________________________________________________________
Added Feb. 17, 2019.
Problem Chapter 4.2.4.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x(x^n + (a n -1) y^n) w_x + y(y^n + (a n -1) x^n) w_y = k n (x^n + y^n) w \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; pde = x*(x^n + (a*n - 1)*y^n)*D[w[x, y], x] + y*(y^n + (a*n - 1)*x^n)*D[w[x, y], y] == k*n*(x^n + y^n)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
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'; g1:='g1';g2:='g2';g0:='g0';h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; pde := x*(x^n + (a*n -1)*y^n)*diff(w(x,y),x)+y*(y^n + (a*n -1)*x^n)*diff(w(x,y),y) = k*n*(x^n + y^n)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( - \left ( {y}^{n}{x}^{-{a}^{-1}}-{x}^{{\frac {an-1}{a}}} \right ) \left ( {y}^{n} \right ) ^{-{\frac {1}{an}}} \right ) {{\rm e}^{\int ^{x}\!{\frac {kn}{{\it \_a}} \left ( {{\it \_a}}^{n}+ \left ( \RootOf \left ( -{y}^{n}{x}^{-{a}^{-1}} \left ( {y}^{n} \right ) ^{-{\frac {1}{an}}}\sqrt [a]{{\it \_a}} \left ( {{\it \_Z}}^{n} \right ) ^{{\frac {1}{an}}}+ \left ( {y}^{n} \right ) ^{-{\frac {1}{an}}}\sqrt [a]{{\it \_a}}{x}^{{\frac {an-1}{a}}} \left ( {{\it \_Z}}^{n} \right ) ^{{\frac {1}{an}}}-{{\it \_a}}^{n}+{{\it \_Z}}^{n} \right ) \right ) ^{n} \right ) \left ( \left ( \RootOf \left ( -{y}^{n}{x}^{-{a}^{-1}} \left ( {y}^{n} \right ) ^{-{\frac {1}{an}}}\sqrt [a]{{\it \_a}} \left ( {{\it \_Z}}^{n} \right ) ^{{\frac {1}{an}}}+ \left ( {y}^{n} \right ) ^{-{\frac {1}{an}}}\sqrt [a]{{\it \_a}}{x}^{{\frac {an-1}{a}}} \left ( {{\it \_Z}}^{n} \right ) ^{{\frac {1}{an}}}-{{\it \_a}}^{n}+{{\it \_Z}}^{n} \right ) \right ) ^{n}an+{{\it \_a}}^{n}- \left ( \RootOf \left ( -{y}^{n}{x}^{-{a}^{-1}} \left ( {y}^{n} \right ) ^{-{\frac {1}{an}}}\sqrt [a]{{\it \_a}} \left ( {{\it \_Z}}^{n} \right ) ^{{\frac {1}{an}}}+ \left ( {y}^{n} \right ) ^{-{\frac {1}{an}}}\sqrt [a]{{\it \_a}}{x}^{{\frac {an-1}{a}}} \left ( {{\it \_Z}}^{n} \right ) ^{{\frac {1}{an}}}-{{\it \_a}}^{n}+{{\it \_Z}}^{n} \right ) \right ) ^{n} \right ) ^{-1}}{d{\it \_a}}}} \]
____________________________________________________________________________________
Added Feb. 17, 2019.
Problem Chapter 4.2.4.13, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x((n-2) y^n -2 x^n ) w_x + y(2 y^n - (n-2) x^n) w_y =\left ( (a(n-2)+2 b) y^n - (2 a + b(n-2)) x^n) \right ) w \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; pde = x*((n - 2)*y^n - 2*x^n)*D[w[x, y], x] + y*(2*y^n - (n - 2)*x^n)*D[w[x, y], y] == ((a*(n - 2) + 2*b)*y^n - (2*a + b*(n - 2))*x^n)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
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';g1:='g1';g2:='g2';g0:='g0'; h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; pde := x*((n-2)*y^n -2*x^n )*diff(w(x,y),x)+y*(2*y^n - (n-2)*x^n)*diff(w(x,y),y) = ((a*(n-2)+2*b)*y^n - (2*a + b*(n-2))*x^n)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { Exception } \]