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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right )+\frac {c \log (\cosh (\lambda x))}{a \lambda }+\frac {k \log (\cosh (\mu y))}{b \mu }\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left ( x,y \right ) ={\frac {1}{2\,a\lambda \,\mu \,b} \left ( 2\,{\it \_F1} \left ( {\frac {ya-xb}{a}} \right ) \mu \,ba\lambda -c\ln \left ( \tanh \left ( x\lambda \right ) -1 \right ) \mu \,b-c\ln \left ( \tanh \left ( x\lambda \right ) +1 \right ) \mu \,b-ak\lambda \, \left ( \ln \left ( \tanh \left ( \mu \,y \right ) -1 \right ) +\ln \left ( \tanh \left ( \mu \,y \right ) +1 \right ) \right ) \right ) }\]
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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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to \frac {c \log (\cosh (\lambda x+\mu y))}{a \lambda +b \mu }+c_1\left (y-\frac {b x}{a}\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left ( x,y \right ) ={\frac {1}{2\,a\lambda +2\,\mu \,b} \left ( \left ( 2\,a\lambda +2\,\mu \,b \right ) {\it \_F1} \left ( {\frac {ya-xb}{a}} \right ) -c \left ( \ln \left ( \tanh \left ( x\lambda +\mu \,y \right ) -1 \right ) +\ln \left ( \tanh \left ( x\lambda +\mu \,y \right ) +1 \right ) \right ) \right ) }\]
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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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to \frac {a x \log (\cosh (\lambda x+\mu y))}{\lambda x+\mu y}+c_1\left (\frac {y}{x}\right )\right \}\right \}\]
Maple ✓
restart; pde :=x*diff(w(x,y),x) + y*diff(w(x,y),y) = a*x*tanh(lambda*x+mu*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left ( x,y \right ) ={\frac {1}{2\,x\lambda +2\,\mu \,y} \left ( 2\,{\it \_F1} \left ( {\frac {y}{x}} \right ) \lambda \,x+2\,{\it \_F1} \left ( {\frac {y}{x}} \right ) \mu \,y-\ln \left ( \tanh \left ( x\lambda +\mu \,y \right ) +1 \right ) ax-\ln \left ( \tanh \left ( x\lambda +\mu \,y \right ) -1 \right ) ax \right ) }\]
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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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to \int _1^x\frac {s \tanh ^k\left (\frac {\beta \left (-b \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\tanh ^2(\lambda x)\right ) \tanh ^{n+1}(\lambda x)+b \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\tanh ^2(\lambda K[1])\right ) \tanh ^{n+1}(\lambda K[1])+a \lambda (n+1) y\right )}{a \lambda (n+1)}\right )+c \tanh ^m(\mu K[1])}{a}dK[1]+c_1\left (y-\frac {b \tanh ^{n+1}(\lambda x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\tanh ^2(\lambda x)\right )}{a \lambda n+a \lambda }\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int \!{\frac {b \left ( \tanh \left ( x\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) +\int ^{x}\!{\frac {1}{a} \left ( c \left ( \tanh \left ( \mu \,{\it \_b} \right ) \right ) ^{m}+s \left ( {\sinh \left ( {\frac {\beta }{a} \left ( \int \! \left ( \tanh \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b}b+ \left ( -\int \!{\frac {b \left ( \tanh \left ( x\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) a \right ) } \right ) \left ( \cosh \left ( {\frac {\beta }{a} \left ( \int \! \left ( \tanh \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b}b+ \left ( -\int \!{\frac {b \left ( \tanh \left ( x\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) a \right ) } \right ) \right ) ^{-1}} \right ) ^{k} \right ) }{d{\it \_b}}\]
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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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to \int _1^y\frac {\tanh ^{-n}(\lambda K[1]) \left (s \tanh ^k(\beta K[1])+c \tanh ^m\left (\frac {-a \mu \, _2F_1\left (1,\frac {1}{2}-\frac {n}{2};\frac {3}{2}-\frac {n}{2};\tanh ^2(\lambda y)\right ) \tanh ^{1-n}(\lambda y)+a \mu \, _2F_1\left (1,\frac {1}{2}-\frac {n}{2};\frac {3}{2}-\frac {n}{2};\tanh ^2(\lambda K[1])\right ) \tanh ^{1-n}(\lambda K[1])+b \lambda \mu x-b \lambda \mu n x}{b \lambda -b \lambda n}\right )\right )}{b}dK[1]+c_1\left (\frac {\tanh ^{1-n}(\lambda y) \, _2F_1\left (1,\frac {1}{2}-\frac {n}{2};\frac {3}{2}-\frac {n}{2};\tanh ^2(\lambda y)\right )}{\lambda -\lambda n}-\frac {b x}{a}\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {a\int \! \left ( \tanh \left ( \lambda \,y \right ) \right ) ^{-n}\,{\rm d}y}{b}}+x \right ) +\int ^{y}\!{\frac { \left ( \tanh \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{b} \left ( s \left ( \tanh \left ( \beta \,{\it \_b} \right ) \right ) ^{k}+ \left ( -{\sinh \left ( {\frac {\mu \, \left ( a\int \! \left ( \tanh \left ( \lambda \,y \right ) \right ) ^{-n}\,{\rm d}y-a\int \! \left ( \tanh \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_b}-xb \right ) }{b}} \right ) \left ( \cosh \left ( {\frac {\mu \, \left ( a\int \! \left ( \tanh \left ( \lambda \,y \right ) \right ) ^{-n}\,{\rm d}y-a\int \! \left ( \tanh \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_b}-xb \right ) }{b}} \right ) \right ) ^{-1}} \right ) ^{m}c \right ) }{d{\it \_b}}\]
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