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'));
 

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

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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'));
 

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

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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'));
 

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

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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'));
 

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

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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'));
 

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

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