6.3.12 4.5

6.3.12.1 [919] Problem 1

problem number 919

Added Feb. 11, 2019.

Problem Chapter 3.4.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b w_y = c \sinh (\lambda x) + k \cosh (\mu y) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == c*Sinh[lambda*x] + k*Cosh[mu*y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right )+\frac {c \cosh (\lambda x)}{a \lambda }+\frac {k \sinh (\mu y)}{b \mu }\right \}\right \}\]

Maple ✓

restart; 
pde :=a*diff(w(x,y),x) + b*diff(w(x,y),y) =  c*sinh(lambda*x)+ k*cosh(mu*y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left ( x,y \right ) ={\frac {1}{\mu \,ba\lambda } \left ( {\it \_F1} \left ( {\frac {ya-xb}{a}} \right ) \mu \,ba\lambda +ka\sinh \left ( \mu \,y \right ) \lambda +\cosh \left ( x\lambda \right ) c\mu \,b \right ) }\]

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6.3.12.2 [920] Problem 2

problem number 920

Added Feb. 11, 2019.

Problem Chapter 3.4.5.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b w_y = \tanh (\lambda x) + k \coth (\mu y) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == Tanh[lambda*x] + k*Coth[mu*y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right )+\frac {\log (\cosh (\lambda x))}{a \lambda }+\frac {k (\log (\tanh (\mu y))+\log (\cosh (\mu y)))}{b \mu }\right \}\right \}\]

Maple ✓

restart; 
pde :=a*diff(w(x,y),x) + b*diff(w(x,y),y) =  tanh(lambda*x)+ k*coth(mu*y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left ( x,y \right ) ={\frac {\ln \left ( \cosh \left ( x\lambda \right ) \right ) }{a\lambda }}+{\frac {k\ln \left ( \sinh \left ( \mu \,y \right ) \right ) }{\mu \,b}}+{\it \_F1} \left ( {\frac {ya-xb}{a}} \right ) \]

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6.3.12.3 [921] Problem 3

problem number 921

Added Feb. 11, 2019.

Problem Chapter 3.4.5.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b w_y = \sinh (\lambda x) + k \tanh (\mu y) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == Sinh[lambda*x] + k*Tanh[mu*y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right )+\frac {\cosh (\lambda x)}{a \lambda }+\frac {k \log (\cosh (\mu y))}{b \mu }\right \}\right \}\]

Maple ✓

restart; 
pde :=a*diff(w(x,y),x) + b*diff(w(x,y),y) =  sinh(lambda*x)+ k*tanh(mu*y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left ( x,y \right ) ={\frac {k\ln \left ( \cosh \left ( \mu \,y \right ) \right ) }{\mu \,b}}+{\frac {\cosh \left ( x\lambda \right ) }{a\lambda }}+{\it \_F1} \left ( {\frac {ya-xb}{a}} \right ) \]

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6.3.12.4 [922] Problem 4

problem number 922

Added Feb. 11, 2019.

Problem Chapter 3.4.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b \cosh (\mu y)w_y = \sinh (\lambda x) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*Cosh[mu*y]*D[w[x, y], y] == Sinh[lambda*x]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \frac {\cosh (\lambda x)}{a \lambda }+c_1\left (\frac {2 \tan ^{-1}\left (\tanh \left (\frac {\mu y}{2}\right )\right )}{\mu }-\frac {b x}{a}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=a*diff(w(x,y),x) + b*cosh(mu*y)*diff(w(x,y),y) =  sinh(lambda*x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left ( x,y \right ) ={\frac {1}{a\lambda } \left ( {\it \_F1} \left ( {\frac {-b\mu \,x+2\,\arctan \left ( {{\rm e}^{\mu \,y}} \right ) a}{\mu \,b}} \right ) a\lambda +\cosh \left ( x\lambda \right ) \right ) }\]

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6.3.12.5 [923] Problem 5

problem number 923

Added Feb. 11, 2019.

Problem Chapter 3.4.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b \sinh (\mu y)w_y = \cosh (\lambda x) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*Sinh[mu*y]*D[w[x, y], y] == Cosh[lambda*x]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \frac {\sinh (\lambda x)}{a \lambda }+c_1\left (\frac {\log \left (\tanh \left (\frac {\mu y}{2}\right )\right )}{\mu }-\frac {b x}{a}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=a*diff(w(x,y),x) + b*sinh(mu*y)*diff(w(x,y),y) =  cosh(lambda*x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left ( x,y \right ) ={\frac {1}{a\lambda } \left ( {\it \_F1} \left ( {\frac {-b\mu \,x-2\,\arctanh \left ( {{\rm e}^{\mu \,y}} \right ) a}{\mu \,b}} \right ) a\lambda +\sinh \left ( x\lambda \right ) \right ) }\]

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