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Added Feb. 23, 2019.
Problem Chapter 4.4.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = (c \tanh (\lambda x) + k \tanh (\mu y)) w \]
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*Tanh[lambda*x] + k*Tanh[mu*y])*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \cosh ^{\frac {c}{a \lambda }}(\lambda x) \cosh ^{\frac {k}{b \mu }}(\mu y)\right \}\right \} \]
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';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*tanh(lambda*x) + k*tanh(mu*y))*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \left ( \tanh \left ( \lambda \,x \right ) -1 \right ) ^{-1/2\,{\frac {c}{a\lambda }}} \left ( \tanh \left ( \lambda \,x \right ) +1 \right ) ^{-1/2\,{\frac {c}{a\lambda }}} \left ( \tanh \left ( \mu \,y \right ) -1 \right ) ^{-1/2\,{\frac {k}{b\mu }}} \left ( \tanh \left ( \mu \,y \right ) +1 \right ) ^{-1/2\,{\frac {k}{b\mu }}} \]
____________________________________________________________________________________
Added Feb. 23, 2019.
Problem Chapter 4.4.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c \tanh (\lambda x +\mu y) w \]
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*Tanh[lambda*x + mu*y]*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \cosh ^{\frac {c}{a \lambda +b \mu }}(\lambda x+\mu y)\right \}\right \} \]
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';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*tanh(lambda*x+mu*y)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); sol:=simplify(sol);
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \left ( \tanh \left ( \lambda \,x+\mu \,y \right ) -1 \right ) ^{-{\frac {c}{2\,a\lambda +2\,b\mu }}} \left ( \tanh \left ( \lambda \,x+\mu \,y \right ) +1 \right ) ^{-{\frac {c}{2\,a\lambda +2\,b\mu }}} \]
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Added Feb. 23, 2019.
Problem Chapter 4.4.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + y w_y = a x \tanh (\lambda x +\mu y) w \]
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*Tanh[lambda*x + mu*y]*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {y}{x}\right ) \cosh ^{\frac {a}{\lambda +\frac {\mu y}{x}}}\left (x \left (\lambda +\frac {\mu y}{x}\right )\right )\right \}\right \} \]
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 := x*diff(w(x,y),x)+y*diff(w(x,y),y) = a*x*tanh(lambda*x+mu*y)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {y}{x}} \right ) \left ( \tanh \left ( \lambda \,x+\mu \,y \right ) -1 \right ) ^{-1/2\,{a \left ( {\frac {\mu \,y}{x}}+\lambda \right ) ^{-1}}} \left ( \tanh \left ( \lambda \,x+\mu \,y \right ) +1 \right ) ^{-1/2\,{a \left ( {\frac {\mu \,y}{x}}+\lambda \right ) ^{-1}}} \]
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Added Feb. 23, 2019.
Problem Chapter 4.4.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \tanh ^n(\lambda x) w_y = (c \tanh ^m(\mu x)+s \tanh ^k(\beta y)) w \]
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*Tanh[lambda*x]^n*D[w[x, y], y] == (c*Tanh[mu*x]^m + s*Tanh[beta*y]^k)*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*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)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \!{\frac {b \left ( \tanh \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{a} \left ( c \left ( \tanh \left ( {\it \_b}\,\mu \right ) \right ) ^{m}+s \left ( \tanh \left ( \beta \,\int \!{\frac {b \left ( \tanh \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}{\it \_b}+ \left ( -\int \!{\frac {b \left ( \tanh \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) \beta \right ) \right ) ^{k} \right ) }{d{\it \_b}}}} \]
____________________________________________________________________________________
Added Feb. 23, 2019.
Problem Chapter 4.4.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \tanh ^n(\lambda y) w_y = (c \tanh ^m(\mu x)+s \tanh ^k(\beta y)) w \]
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*Tanh[lambda*y]^n*D[w[x, y], y] == (c*Tanh[mu*x]^m + s*Tanh[beta*y]^k)*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*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)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {a\int \! \left ( \tanh \left ( y\lambda \right ) \right ) ^{-n}\,{\rm d}y}{b}}+x \right ) {{\rm e}^{\int ^{y}\!{\frac { \left ( \tanh \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{b} \left ( c \left ( -\tanh \left ( -\mu \,\int \!{\frac { \left ( \tanh \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}a}{b}}\,{\rm d}{\it \_b}-\mu \, \left ( -{\frac {a\int \! \left ( \tanh \left ( y\lambda \right ) \right ) ^{-n}\,{\rm d}y}{b}}+x \right ) \right ) \right ) ^{m}+s \left ( \tanh \left ( \beta \,{\it \_b} \right ) \right ) ^{k} \right ) }{d{\it \_b}}}} \]