6.8.11 4.5

6.8.11.1 [1827] Problem 1

problem number 1827

Added Oct 10, 2019.

Problem Chapter 8.4.5.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ a w_x + b w_y + c \sinh ^n(\lambda y) w_z = \left ( s \cosh ^m(\beta x)+k \sinh ^r(\gamma y) \right ) w \]

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)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},z-\frac {c \sqrt {\cosh ^2(\lambda y)} \text {sech}(\lambda y) \sinh ^{n+1}(\lambda y) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};-\sinh ^2(\lambda y)\right )}{b \lambda n+b \lambda }\right ) \exp \left (\frac {s \sqrt {-\sinh ^2(\beta x)} \text {csch}(\beta x) \cosh ^{m+1}(\beta x) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cosh ^2(\beta x)\right )}{a \beta m+a \beta }+\frac {k \sqrt {\cosh ^2(\gamma y)} \text {sech}(\gamma y) \sinh ^{r+1}(\gamma y) \, _2F_1\left (\frac {1}{2},\frac {r+1}{2};\frac {r+3}{2};-\sinh ^2(\gamma y)\right )}{b \gamma r+b \gamma }\right )\right \}\right \}\]

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)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ay-xb}{a}},-\int ^{x}\!{\frac {c}{a} \left ( \sinh \left ( {\frac {\lambda \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n}}{d{\it \_a}}+z \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{a} \left ( s \left ( \cosh \left ( \beta \,{\it \_a} \right ) \right ) ^{m}+k \left ( \sinh \left ( {\frac {\gamma \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{r} \right ) }{d{\it \_a}}}}\]

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6.8.11.2 [1828] Problem 2

problem number 1828

Added Oct 10, 2019.

Problem Chapter 8.4.5.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {s \sqrt {-\sinh ^2(\gamma x)} \text {csch}(\gamma x) \cosh ^{k+1}(\gamma x) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};\cosh ^2(\gamma x)\right )}{\gamma k+\gamma }\right ) c_1\left (\frac {b \sinh (\beta x) \cosh ^{m+1}(\beta x) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cosh ^2(\beta x)\right )}{(\beta m+\beta ) \sqrt {-\sinh ^2(\beta x)}}+z,y-\frac {a \sqrt {\cosh ^2(\lambda x)} \text {sech}(\lambda x) \sinh ^{n+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};-\sinh ^2(\lambda x)\right )}{\lambda n+\lambda }\right )\right \}\right \}\]

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)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!a \left ( \sinh \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( \cosh \left ( \beta \,x \right ) \right ) ^{m}\,{\rm d}x+z \right ) {{\rm e}^{\int \!s \left ( \cosh \left ( x\gamma \right ) \right ) ^{k}\,{\rm d}x}}\]

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6.8.11.3 [1829] Problem 3

problem number 1829

Added Oct 10, 2019.

Problem Chapter 8.4.5.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✗

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

Failed

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)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!a \left ( \cosh \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y,-\int ^{x}\!b \left ( \sinh \left ( \beta \, \left ( \int \! \left ( \cosh \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b}a-\int \!a \left ( \cosh \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{m}{d{\it \_b}}+z \right ) {{\rm e}^{\int ^{x}\!s \left ( \sinh \left ( \gamma \, \left ( \int \!b \left ( \sinh \left ( \beta \, \left ( a\int \! \left ( \cosh \left ( {\it \_f}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_f}-\int \!a \left ( \cosh \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{m}\,{\rm d}{\it \_f}-\int ^{x}\!b \left ( \sinh \left ( \beta \, \left ( \int \! \left ( \cosh \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b}a-\int \!a \left ( \cosh \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{m}{d{\it \_b}}+z \right ) \right ) \right ) ^{k}{d{\it \_f}}}}\]

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6.8.11.4 [1830] Problem 4

problem number 1830

Added Oct 10, 2019.

Problem Chapter 8.4.5.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + a \tanh ^n(\lambda x) w_y + b \coth ^m(\beta x) w_z = s \coth ^k(\gamma x) w \]

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*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {s \coth ^{k+1}(\gamma x) \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};\coth ^2(\gamma x)\right )}{\gamma k+\gamma }\right ) c_1\left (z-\frac {b \coth ^{m+1}(\beta x) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};\coth ^2(\beta x)\right )}{\beta m+\beta },y-\frac {a \tanh ^{n+1}(\lambda x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\tanh ^2(\lambda x)\right )}{\lambda n+\lambda }\right )\right \}\right \}\]

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)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!a \left ( \tanh \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( {\rm coth} \left (\beta \,x\right ) \right ) ^{m}\,{\rm d}x+z \right ) {{\rm e}^{\int \!s \left ( {\rm coth} \left (x\gamma \right ) \right ) ^{k}\,{\rm d}x}}\]

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6.8.11.5 [1831] Problem 5

problem number 1831

Added Oct 10, 2019.

Problem Chapter 8.4.5.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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]*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \sinh ^{\frac {k}{a \lambda }}(\lambda x) c_1\left (\frac {\log \left (\tanh \left (\frac {\beta y}{2}\right ) \tanh ^{-\frac {b \beta }{a \lambda }}\left (\frac {\lambda x}{2}\right )\right )}{\beta },\frac {\log \left (\tanh \left (\frac {\gamma z}{2}\right ) \tanh ^{-\frac {c \gamma }{a \lambda }}\left (\frac {\lambda x}{2}\right )\right )}{\gamma }\right )\right \}\right \}\]

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)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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