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'));
 

\[w \left (x , y\right ) = \frac {f_{1} \left (\frac {y a -b x}{a}\right ) \mu b a \lambda +k a \cosh \left (\mu y \right ) \lambda +\cosh \left (\lambda x \right ) c \mu b}{\mu b a \lambda }\]

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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'));
 

\[w \left (x , y\right ) = \frac {c \cosh \left (\lambda x +\mu y \right )}{\lambda a +\mu b}+f_{1} \left (\frac {a y -b x}{a}\right )\]

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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]];
 

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

Maple ✓

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

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

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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]];
 

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

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'));
 

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

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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]];
 

\begin{align*}& \left \{w(x,y)\to \int _1^x\frac {s \left (-\sinh \left (\frac {\beta \text {arccosh}\left (\tanh \left (\text {arctanh}(\cosh (\lambda y))+\frac {b \lambda (x-K[1])}{a}\right )\right )}{\lambda }\right )\right )^k+c \sinh ^m(\mu K[1])}{a}dK[1]+c_1\left (-\frac {b x}{a}-\frac {\text {arctanh}(\cosh (\lambda y))}{\lambda }\right )\right \}\\& \left \{w(x,y)\to \int _1^x\frac {s \sinh ^k\left (\frac {\beta \text {arccosh}\left (\tanh \left (\text {arctanh}(\cosh (\lambda y))+\frac {b \lambda (x-K[2])}{a}\right )\right )}{\lambda }\right )+c \sinh ^m(\mu K[2])}{a}dK[2]+c_1\left (-\frac {b x}{a}-\frac {\text {arctanh}(\cosh (\lambda y))}{\lambda }\right )\right \}\\\end{align*}

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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-x b \lambda -2 \,\operatorname {arctanh}\left ({\mathrm e}^{\lambda y}\right ) a}{b \lambda }\right )+\frac {\int _{}^{x}\left ({\sinh \left (\frac {\beta \ln \left (\tanh \left (\frac {2 \,\operatorname {arctanh}\left ({\mathrm e}^{\lambda y}\right ) a +b \lambda \left (x -\textit {\_a} \right )}{2 a}\right )\right )}{\lambda }\right )}^{k} s +c \sinh \left (\mu \textit {\_a} \right )^{m}\right )d \textit {\_a}}{a}\]

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