6.3.8 4.1

6.3.8.1 [899] Problem 1

problem number 899

Added Feb. 9, 2019.

Problem Chapter 3.4.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.3.8.2 [900] Problem 2

problem number 900

Added Feb. 9, 2019.

Problem Chapter 3.4.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) ={\frac {c\cosh \left ( \lambda \,x+\mu \,y \right ) }{a\lambda +\mu \,b}}+{\it \_F1} \left ( {\frac {ay-bx}{a}} \right ) \]

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6.3.8.3 [901] Problem 3

problem number 901

Added Feb. 9, 2019.

Problem Chapter 3.4.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

$Aborted

Maple ✓

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

\[w \left ( x,y \right ) ={\frac {1}{ \left ( a\lambda +\mu \,b \right ) ^{2}} \left ( \left ( a\lambda +\mu \,b \right ) ^{2}{\it \_F1} \left ( {\frac {ay-bx}{a}} \right ) - \left ( -x \left ( a\lambda +\mu \,b \right ) \cosh \left ( \lambda \,x+\mu \,y \right ) +a\sinh \left ( \lambda \,x+\mu \,y \right ) \right ) c \right ) }\]

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6.3.8.4 [902] Problem 4

problem number 902

Added Feb. 9, 2019.

Problem Chapter 3.4.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \sinh ^n(\lambda x) w_y = c \sinh ^m(\mu x)+s \sinh ^k(\beta y) \]

Mathematica ✗

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

$Aborted Kernel Exception

Maple ✓

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

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {y\lambda \,a-b\cosh \left ( \lambda \,x \right ) }{a\lambda }} \right ) +\int ^{x}\!{\frac {1}{a} \left ( c \left ( \sinh \left ( \mu \,{\it \_a} \right ) \right ) ^{m}+s \left ( \sinh \left ( {\frac {\beta \, \left ( y\lambda \,a-b\cosh \left ( \lambda \,x \right ) +b\cosh \left ( {\it \_a}\,\lambda \right ) \right ) }{a\lambda }} \right ) \right ) ^{k} \right ) }{d{\it \_a}}\]

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6.3.8.5 [903] Problem 5

problem number 903

Added Feb. 9, 2019.

Problem Chapter 3.4.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \sinh ^n(\lambda y) w_y = c \sinh ^m(\mu x)+s \sinh ^k(\beta y) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \int _1^x\frac {s \sinh ^k\left (\frac {2 \beta \tanh ^{-1}\left (e^{\frac {b \lambda (K[1]-x)}{a}} \tanh \left (\frac {\lambda y}{2}\right )\right )}{\lambda }\right )+c \sinh ^m(\mu K[1])}{a}dK[1]+c_1\left (\frac {\log \left (\tanh \left (\frac {\lambda y}{2}\right )\right )}{\lambda }-\frac {b x}{a}\right )\right \}\right \}\]

Maple ✓

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

\[w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{a} \left ( c \left ( \sinh \left ( \mu \,{\it \_a} \right ) \right ) ^{m}+s \left ( \sinh \left ( {\frac {\beta }{\lambda }\ln \left ( \tanh \left ( {\frac {1}{a} \left ( \arctanh \left ( {{\rm e}^{\lambda \,y}} \right ) a+{\frac {b\lambda \, \left ( x-{\it \_a} \right ) }{2}} \right ) } \right ) \right ) } \right ) \right ) ^{k} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {-xb\lambda -2\,\arctanh \left ( {{\rm e}^{\lambda \,y}} \right ) a}{b\lambda }} \right ) \]

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