6.7.11 4.5

6.7.11.1 [1658] Problem 1

problem number 1658

Added June 20, 2019.

Problem Chapter 7.4.5.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},z-\frac {c \sqrt {\cosh ^2(\lambda y)} \text {sech}(\lambda y) \sinh ^{n+1}(\lambda y) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},-\sinh ^2(\lambda y)\right )}{b \lambda n+b \lambda }\right )+\frac {s \sqrt {-\sinh ^2(\beta x)} \text {csch}(\beta x) \cosh ^{m+1}(\beta x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cosh ^2(\beta x)\right )}{a \beta m+a \beta }+\frac {k \sqrt {\cosh ^2(\gamma y)} \text {sech}(\gamma y) \sinh ^{r+1}(\gamma y) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {r+1}{2},\frac {r+3}{2},-\sinh ^2(\gamma y)\right )}{b \gamma r+b \gamma }\right \}\right \}\]

Maple ✓

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

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

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6.7.11.2 [1659] Problem 2

problem number 1659

Added June 20, 2019.

Problem Chapter 7.4.5.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a \sinh ^n(\lambda x) w_y + b \cosh ^m(\beta x) w_z = s \cosh ^k(\gamma x) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to \frac {s \sqrt {-\sinh ^2(\gamma x)} \text {csch}(\gamma x) \cosh ^{k+1}(\gamma x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {k+1}{2},\frac {k+3}{2},\cosh ^2(\gamma x)\right )}{\gamma k+\gamma }+c_1\left (\frac {b \sinh (\beta x) \cosh ^{m+1}(\beta x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cosh ^2(\beta x)\right )}{(\beta m+\beta ) \sqrt {-\sinh ^2(\beta x)}}+z,y-\frac {a \sqrt {\cosh ^2(\lambda x)} \text {sech}(\lambda x) \sinh ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},-\sinh ^2(\lambda x)\right )}{\lambda n+\lambda }\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y , z\right ) = s \int \cosh \left (\gamma x \right )^{k}d x +f_{1} \left (-a \int \sinh \left (\lambda x \right )^{n}d x +y , -b \int \cosh \left (\beta x \right )^{m}d x +z \right )\]

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6.7.11.3 [1660] Problem 3

problem number 1660

Added June 20, 2019.

Problem Chapter 7.4.5.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a \cosh ^n(\lambda x) w_y + b \sinh ^m(\beta y) w_z = s \sinh ^k(\gamma z) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to \int _1^xs \sinh ^k\left (\frac {\gamma \left (-b \sqrt {\cosh ^2(\beta x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},-\sinh ^2(\beta x)\right ) \text {sech}(\beta x) \sinh ^{m+1}(\beta x)+b \sqrt {\cosh ^2(\beta K[1])} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},-\sinh ^2(\beta K[1])\right ) \text {sech}(\beta K[1]) \sinh ^{m+1}(\beta K[1])+\beta (m+1) z\right )}{\beta (m+1)}\right )dK[1]+c_1\left (z-\frac {b \sqrt {\cosh ^2(\beta x)} \text {sech}(\beta x) \sinh ^{m+1}(\beta x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},-\sinh ^2(\beta x)\right )}{\beta m+\beta },\frac {a \sinh (\lambda x) \cosh ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cosh ^2(\lambda x)\right )}{(\lambda n+\lambda ) \sqrt {-\sinh ^2(\lambda x)}}+y\right )\right \}\right \}\]

Maple ✓

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

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

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6.7.11.4 [1661] Problem 4

problem number 1661

Added June 20, 2019.

Problem Chapter 7.4.5.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a \tanh ^n(\lambda x) w_y + b \coth ^m(\beta x) w_z = s \coth ^k(\gamma x) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to \frac {s \coth ^{k+1}(\gamma x) \operatorname {Hypergeometric2F1}\left (1,\frac {k+1}{2},\frac {k+3}{2},\coth ^2(\gamma x)\right )}{\gamma k+\gamma }+c_1\left (z-\frac {b \coth ^{m+1}(\beta x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},\coth ^2(\beta x)\right )}{\beta m+\beta },y-\frac {a \tanh ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\tanh ^2(\lambda x)\right )}{\lambda n+\lambda }\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y , z\right ) = s \int \coth \left (\gamma x \right )^{k}d x +f_{1} \left (-a \int \tanh \left (\lambda x \right )^{n}d x +y , -b \int \coth \left (\beta x \right )^{m}d x +z \right )\]

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6.7.11.5 [1662] Problem 5

problem number 1662

Added June 20, 2019.

Problem Chapter 7.4.5.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a \sinh (\lambda x) w_x + b \sinh (\beta y) w_y + c \sinh (\gamma z) w_z = k \cosh (\lambda x) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to \frac {k \log (\sinh (\lambda x))}{a \lambda }+c_1\left (\frac {b \text {arctanh}(\cosh (\lambda x))}{a \lambda }-\frac {\text {arctanh}(\cosh (\beta y))}{\beta },\frac {c \text {arctanh}(\cosh (\lambda x))}{a \lambda }-\frac {\text {arctanh}(\cosh (\gamma z))}{\gamma }\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y , z\right ) = \frac {f_{1} \left (\frac {-\ln \left (\tanh \left (\frac {\lambda x}{2}\right )\right ) b \beta -2 a \,\operatorname {arctanh}\left ({\mathrm e}^{\beta y}\right ) \lambda }{\lambda b \beta }, \frac {-\ln \left (\tanh \left (\frac {\lambda x}{2}\right )\right ) c \gamma -2 a \,\operatorname {arctanh}\left ({\mathrm e}^{\gamma z}\right ) \lambda }{\lambda c \gamma }\right ) a \lambda +k \ln \left (\sinh \left (\lambda x \right )\right )}{a \lambda }\]

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