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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},z-\frac {c \sqrt {\cosh ^2(\lambda y)} \text {sech}(\lambda y) \sinh ^{n+1}(\lambda y) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},-\sinh ^2(\lambda y)\right )}{b \lambda n+b \lambda }\right ) \exp \left (\frac {s \sqrt {-\sinh ^2(\beta x)} \text {csch}(\beta x) \cosh ^{m+1}(\beta x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cosh ^2(\beta x)\right )}{a \beta m+a \beta }+\frac {k \sqrt {\cosh ^2(\gamma y)} \text {sech}(\gamma y) \sinh ^{r+1}(\gamma y) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {r+1}{2},\frac {r+3}{2},-\sinh ^2(\gamma y)\right )}{b \gamma r+b \gamma }\right )\right \}\right \}\]
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'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {a y -b x}{a}, -\frac {c \int _{}^{x}\sinh \left (\frac {\left (a y -b \left (x -\textit {\_a} \right )\right ) \lambda }{a}\right )^{n}d \textit {\_a}}{a}+z \right ) {\mathrm e}^{\frac {\int _{}^{x}\left (s \cosh \left (\beta \textit {\_a} \right )^{m}+k \sinh \left (\frac {\left (a y -b \left (x -\textit {\_a} \right )\right ) \gamma }{a}\right )^{r}\right )d \textit {\_a}}{a}}\]
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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]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {s \sqrt {-\sinh ^2(\gamma x)} \text {csch}(\gamma x) \cosh ^{k+1}(\gamma x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {k+1}{2},\frac {k+3}{2},\cosh ^2(\gamma x)\right )}{\gamma k+\gamma }\right ) c_1\left (\frac {b \sinh (\beta x) \cosh ^{m+1}(\beta x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cosh ^2(\beta x)\right )}{(\beta m+\beta ) \sqrt {-\sinh ^2(\beta x)}}+z,y-\frac {a \sqrt {\cosh ^2(\lambda x)} \text {sech}(\lambda x) \sinh ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},-\sinh ^2(\lambda x)\right )}{\lambda n+\lambda }\right )\right \}\right \}\]
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)*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (-a \int \sinh \left (\lambda x \right )^{n}d x +y , -b \int \cosh \left (\beta x \right )^{m}d x +z \right ) {\mathrm e}^{s \int \cosh \left (\gamma x \right )^{k}d x}\]
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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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {a \sinh (\lambda x) \cosh ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cosh ^2(\lambda x)\right )}{(\lambda n+\lambda ) \sqrt {-\sinh ^2(\lambda x)}}+y,z-\int _1^xb \sinh ^m\left (\frac {\beta \left (\frac {a \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cosh ^2(\lambda x)\right ) \sinh (\lambda x) \cosh ^{n+1}(\lambda x)}{\sqrt {-\sinh ^2(\lambda x)}}+\lambda (n+1) y+a \cosh ^{n+1}(\lambda K[1]) \text {csch}(\lambda K[1]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cosh ^2(\lambda K[1])\right ) \sqrt {-\sinh ^2(\lambda K[1])}\right )}{\lambda (n+1)}\right )dK[1]\right ) \exp \left (\int _1^xs \sinh ^k\left (\gamma \left (z-\int _1^xb \sinh ^m\left (\frac {\beta \left (\frac {a \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cosh ^2(\lambda x)\right ) \sinh (\lambda x) \cosh ^{n+1}(\lambda x)}{\sqrt {-\sinh ^2(\lambda x)}}+\lambda (n+1) y+a \cosh ^{n+1}(\lambda K[1]) \text {csch}(\lambda K[1]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cosh ^2(\lambda K[1])\right ) \sqrt {-\sinh ^2(\lambda K[1])}\right )}{\lambda (n+1)}\right )dK[1]+\int _1^{K[2]}b \sinh ^m\left (\frac {\beta \left (\frac {a \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cosh ^2(\lambda x)\right ) \sinh (\lambda x) \cosh ^{n+1}(\lambda x)}{\sqrt {-\sinh ^2(\lambda x)}}+\lambda (n+1) y+a \cosh ^{n+1}(\lambda K[1]) \text {csch}(\lambda K[1]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cosh ^2(\lambda K[1])\right ) \sqrt {-\sinh ^2(\lambda K[1])}\right )}{\lambda (n+1)}\right )dK[1]\right )\right )dK[2]\right )\right \}\right \}\]
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)*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (-a \int \cosh \left (\lambda x \right )^{n}d x +y , -b \int _{}^{x}{\left (-\sinh \left (\beta \left (a \int \cosh \left (\lambda x \right )^{n}d x -a \int \cosh \left (\lambda \textit {\_f} \right )^{n}d \textit {\_f} -y \right )\right )\right )}^{m}d \textit {\_f} +z \right ) {\mathrm e}^{s \int _{}^{x}{\sinh \left (\gamma \left (-b \int _{}^{x}{\left (-\sinh \left (\beta \left (a \int \cosh \left (\lambda x \right )^{n}d x -a \int \cosh \left (\lambda \textit {\_f} \right )^{n}d \textit {\_f} -y \right )\right )\right )}^{m}d \textit {\_f} +b \int {\left (-\sinh \left (\beta \left (a \int \cosh \left (\lambda x \right )^{n}d x -a \int \cosh \left (\lambda \textit {\_f} \right )^{n}d \textit {\_f} -y \right )\right )\right )}^{m}d \textit {\_f} +z \right )\right )}^{k}d \textit {\_f}}\]
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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]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {s \coth ^{k+1}(\gamma x) \operatorname {Hypergeometric2F1}\left (1,\frac {k+1}{2},\frac {k+3}{2},\coth ^2(\gamma x)\right )}{\gamma k+\gamma }\right ) c_1\left (z-\frac {b \coth ^{m+1}(\beta x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},\coth ^2(\beta x)\right )}{\beta m+\beta },y-\frac {a \tanh ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\tanh ^2(\lambda x)\right )}{\lambda n+\lambda }\right )\right \}\right \}\]
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)*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (-a \int \tanh \left (\lambda x \right )^{n}d x +y , -b \int \coth \left (\beta x \right )^{m}d x +z \right ) {\mathrm e}^{s \int \coth \left (\gamma x \right )^{k}d x}\]
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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]];
\[\left \{\left \{w(x,y,z)\to \sinh ^{\frac {k}{a \lambda }}(\lambda x) c_1\left (\frac {b \text {arctanh}(\cosh (\lambda x))}{a \lambda }-\frac {\text {arctanh}(\cosh (\beta y))}{\beta },\frac {c \text {arctanh}(\cosh (\lambda x))}{a \lambda }-\frac {\text {arctanh}(\cosh (\gamma z))}{\gamma }\right )\right \}\right \}\]
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)*w(x,y,z);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = f_{1} \left (\frac {-\ln \left (\tanh \left (\frac {\lambda x}{2}\right )\right ) b \beta -2 a \,\operatorname {arctanh}\left ({\mathrm e}^{\beta y}\right ) \lambda }{\lambda b \beta }, \frac {-\ln \left (\tanh \left (\frac {\lambda x}{2}\right )\right ) c \gamma -2 a \,\operatorname {arctanh}\left ({\mathrm e}^{\gamma z}\right ) \lambda }{\lambda c \gamma }\right ) \sinh \left (\lambda x \right )^{\frac {k}{a \lambda }}\]
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