____________________________________________________________________________________
Added Feb. 9, 2019.
Problem Chapter 3.4.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c \tanh (\lambda x)+ k \tanh (\mu y) \]
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, q, p, delta]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*Tanh[lambda*x] + k*Tanh[mu*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {a b \lambda \mu c_1\left (\frac {a y-b x}{a}\right )+a k \lambda \log \left (\cosh \left (\frac {\mu (a y-b x)}{a}+\frac {b \mu x}{a}\right )\right )+b c \mu \log (\cosh (\lambda x))}{a b \lambda \mu }\right \}\right \} \]
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*diff(w(x,y),x) + b*diff(w(x,y),y) = c*tanh(lambda*x)+ k *tanh(mu*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) =1/2\,{\frac {1}{b\mu \,a\lambda } \left ( 2\,{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) b\mu \,a\lambda -k\ln \left ( \tanh \left ( \mu \,y \right ) -1 \right ) a\lambda -k\ln \left ( \tanh \left ( \mu \,y \right ) +1 \right ) a\lambda -c\ln \left ( \tanh \left ( \lambda \,x \right ) -1 \right ) b\mu -c\ln \left ( \tanh \left ( \lambda \,x \right ) +1 \right ) b\mu \right ) } \]
____________________________________________________________________________________
Added Feb. 9, 2019.
Problem Chapter 3.4.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c \tanh (\lambda x+\mu y) \]
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, q, p, delta]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*Tanh[lambda*x + mu*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {c \log \left (\cosh \left (\frac {x (a \lambda +b \mu )}{a}+\frac {\mu (a y-b x)}{a}\right )\right )+a \lambda c_1\left (\frac {a y-b x}{a}\right )+b \mu c_1\left (\frac {a y-b x}{a}\right )}{a \lambda +b \mu }\right \}\right \} \]
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*diff(w(x,y),x) + b*diff(w(x,y),y) = c*tanh(lambda*x+mu*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) =-1/2\,{\frac {1}{a\lambda +b\mu } \left ( -2\,{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) a\lambda -2\,{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) b\mu +c\ln \left ( \tanh \left ( {\frac { \left ( ya-bx \right ) \mu +ax\lambda +b\mu \,x}{a}} \right ) -1 \right ) +c\ln \left ( \tanh \left ( {\frac { \left ( ya-bx \right ) \mu +ax\lambda +b\mu \,x}{a}} \right ) +1 \right ) \right ) } \]
____________________________________________________________________________________
Added Feb. 11, 2019.
Problem Chapter 3.4.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + y w_y = a x \tanh (\lambda x+\mu y) \]
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, q, p, delta]; pde = x*D[w[x, y], x] + y*D[w[x, y], y] == a*x*Tanh[lambda*x + mu*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {a x \log \left (\cosh \left (x \left (\lambda +\frac {\mu y}{x}\right )\right )\right )+\lambda x c_1\left (\frac {y}{x}\right )+\mu y c_1\left (\frac {y}{x}\right )}{\lambda x+\mu y}\right \}\right \} \]
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 :=x*diff(w(x,y),x) + y*diff(w(x,y),y) = a*x*tanh(lambda*x+mu*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) =-1/2\,{\frac {1}{\lambda \,x+\mu \,y} \left ( \left ( \ln \left ( \tanh \left ( x \left ( {\frac {\mu \,y}{x}}+\lambda \right ) \right ) -1 \right ) a+\ln \left ( \tanh \left ( x \left ( {\frac {\mu \,y}{x}}+\lambda \right ) \right ) +1 \right ) a-2\,{\it \_F1} \left ( {\frac {y}{x}} \right ) \lambda \right ) x-2\,{\it \_F1} \left ( {\frac {y}{x}} \right ) \mu \,y \right ) } \]
____________________________________________________________________________________
Added Feb. 11, 2019.
Problem Chapter 3.4.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \tanh ^n(\lambda x) w_y = c \tanh ^m(\mu x)+s \tanh ^k(\beta y) \]
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, q, p, delta]; pde = a*D[w[x, y], x] + b*Tanh[lambda*x]^n*D[w[x, y], y] == c*Tanh[mu*x]^m + s*Tanh[beta*y]^k; 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*diff(w(x,y),x) + b*tanh(lambda*x)^n*diff(w(x,y),y) = c*tanh(mu*x)^m+s*tanh(beta*y)^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{a} \left ( s \left ( {1\sinh \left ( {\frac {\beta }{a} \left ( \int \! \left ( \tanh \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b}b+ \left ( -\int \!{\frac {b \left ( \tanh \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) a \right ) } \right ) \left ( \cosh \left ( {\frac {\beta }{a} \left ( \int \! \left ( \tanh \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b}b+ \left ( -\int \!{\frac {b \left ( \tanh \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) a \right ) } \right ) \right ) ^{-1}} \right ) ^{k}+c \left ( \tanh \left ( {\it \_b}\,\mu \right ) \right ) ^{m} \right ) }{d{\it \_b}}+{\it \_F1} \left ( -\int \!{\frac {b \left ( \tanh \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) \]
____________________________________________________________________________________
Added Feb. 11, 2019.
Problem Chapter 3.4.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \tanh ^n(\lambda y) w_y = c \tanh ^m(\mu x)+s \tanh ^k(\beta y) \]
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, q, p, delta]; pde = a*D[w[x, y], x] + b*Tanh[lambda*y]^n*D[w[x, y], y] == c*Tanh[mu*x]^m + s*Tanh[beta*y]^k; 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*diff(w(x,y),x) + b*tanh(lambda*y)^n*diff(w(x,y),y) = c*tanh(mu*x)^m+s*tanh(beta*y)^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) =\int ^{y}\!{\frac { \left ( \tanh \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{b} \left ( s \left ( \tanh \left ( \beta \,{\it \_b} \right ) \right ) ^{k}+ \left ( {1\sinh \left ( {\frac {\mu }{b} \left ( a\int \! \left ( \tanh \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_b}+ \left ( -{\frac {a\int \! \left ( \tanh \left ( y\lambda \right ) \right ) ^{-n}\,{\rm d}y}{b}}+x \right ) b \right ) } \right ) \left ( \cosh \left ( {\frac {\mu }{b} \left ( a\int \! \left ( \tanh \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_b}+ \left ( -{\frac {a\int \! \left ( \tanh \left ( y\lambda \right ) \right ) ^{-n}\,{\rm d}y}{b}}+x \right ) b \right ) } \right ) \right ) ^{-1}} \right ) ^{m}c \right ) }{d{\it \_b}}+{\it \_F1} \left ( -{\frac {a\int \! \left ( \tanh \left ( y\lambda \right ) \right ) ^{-n}\,{\rm d}y}{b}}+x \right ) \]