6.9.11 4.5

6.9.11.1 [1991] Problem 1

problem number 1991

Added Jan 19, 2020.

Problem Chapter 9.4.5.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.9.11.2 [1992] Problem 2

problem number 1992

Added Jan 19, 2020.

Problem Chapter 9.4.5.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {c \tanh ^{n+1}(\beta x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\tanh ^2(\beta x)\right )}{\beta n+\beta }\right ) \left (\int _1^x\exp \left (-\frac {c \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\tanh ^2(\beta K[1])\right ) \tanh ^{n+1}(\beta K[1])}{n \beta +\beta }\right ) k \coth ^m(\lambda K[1])dK[1]+c_1(y-a x,z-b x)\right )\right \}\right \}\]

Maple ✓

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

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

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6.9.11.3 [1993] Problem 3

problem number 1993

Added Jan 19, 2020.

Problem Chapter 9.4.5.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.9.11.4 [1994] Problem 4

problem number 1994

Added Jan 19, 2020.

Problem Chapter 9.4.5.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to e^{c x} \left (\int _1^xe^{-c K[1]} s \tanh ^m(\mu K[1])dK[1]+c_1\left (z-\frac {b \coth ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {k+1}{2},\frac {k+3}{2},\coth ^2(\lambda x)\right )}{k \lambda +\lambda },y-\frac {a \tanh ^{n+1}(\beta x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\tanh ^2(\beta x)\right )}{\beta n+\beta }\right )\right )\right \}\right \}\]

Maple ✓

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

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

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6.9.11.5 [1995] Problem 5

problem number 1995

Added Jan 19, 2020.

Problem Chapter 9.4.5.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ b_1 \sinh ^{n_1}(\lambda _1 x) w_x + b_2 \cosh ^{n_2}(\lambda _2 y) w_y + b_3 \sinh ^{n_3}(\lambda _3 z) w_z = a w + c_1 \cosh ^{k_1}(\beta _1 x)+ c_2 \sinh ^{k_2}(\beta _2 y)+ c_3 \sinh ^{k_3}(\beta _3 z) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  b1*Sinh[lambda1*x]^n1*D[w[x,y,z],x]+ b2*Cosh[lambda2*x]^n2*D[w[x,y,z],y]+b3*Sinh[lambda3*x]^n3*D[w[x,y,z],z]==a*w[x,y,z]+ c1*Cosh[beta1*x]^k1+ c2*Sinh[beta2*x]^k2+ c3*Sinh[beta3*x]^k3; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {a \sqrt {\cosh ^2(\text {lambda1} x)} \text {sech}(\text {lambda1} x) \sinh ^{1-\text {n1}}(\text {lambda1} x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-\text {n1}}{2},\frac {3-\text {n1}}{2},-\sinh ^2(\text {lambda1} x)\right )}{\text {b1} \text {lambda1}-\text {b1} \text {lambda1} \text {n1}}\right ) \left (\int _1^x\frac {\exp \left (\frac {a \sqrt {\cosh ^2(\text {lambda1} K[3])} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-\text {n1}}{2},\frac {3-\text {n1}}{2},-\sinh ^2(\text {lambda1} K[3])\right ) \text {sech}(\text {lambda1} K[3]) \sinh ^{1-\text {n1}}(\text {lambda1} K[3])}{\text {b1} \text {lambda1} \text {n1}-\text {b1} \text {lambda1}}\right ) \left (\text {c1} \cosh ^{\text {k1}}(\text {beta1} K[3])+\text {c2} \sinh ^{\text {k2}}(\text {beta2} K[3])+\text {c3} \sinh ^{\text {k3}}(\text {beta3} K[3])\right ) \sinh ^{-\text {n1}}(\text {lambda1} K[3])}{\text {b1}}dK[3]+c_1\left (y-\int _1^x\frac {\text {b2} \cosh ^{\text {n2}}(\text {lambda2} K[1]) \sinh ^{-\text {n1}}(\text {lambda1} K[1])}{\text {b1}}dK[1],z-\int _1^x\frac {\text {b3} \sinh ^{-\text {n1}}(\text {lambda1} K[2]) \sinh ^{\text {n3}}(\text {lambda3} K[2])}{\text {b1}}dK[2]\right )\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde := b__1*sinh(lambda__1*x)^(n__1)*diff(w(x,y,z),x)+b__2*cosh(lambda__2*x)^(n__2)*diff(w(x,y,z),y)+ b__3*sinh(lambda__3*x)^(n__3)*diff(w(x,y,z),z)=a*w(x,y,z)+ c__1*cosh(beta__1*x)^(k__1)+ c__2*sinh(beta__2*x)^(k__2)+ c__3*sinh(beta__3*x)^(k__3); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \left (\frac {\int \left (c_{1} \cosh \left (\beta _{1} x \right )^{k_{1}}+c_{2} \sinh \left (\beta _{2} x \right )^{k_{2}}+c_{3} \sinh \left (\beta _{3} x \right )^{k_{3}}\right ) \sinh \left (\lambda _{1} x \right )^{-n_{1}} {\mathrm e}^{-\frac {a \int \sinh \left (\lambda _{1} x \right )^{-n_{1}}d x}{b_{1}}}d x}{b_{1}}+f_{1} \left (-\frac {b_{2} \int \cosh \left (\lambda _{2} x \right )^{n_{2}} \sinh \left (\lambda _{1} x \right )^{-n_{1}}d x}{b_{1}}+y , -\frac {b_{3} \int \sinh \left (\lambda _{3} x \right )^{n_{3}} \sinh \left (\lambda _{1} x \right )^{-n_{1}}d x}{b_{1}}+z \right )\right ) {\mathrm e}^{\frac {a \int \sinh \left (\lambda _{1} x \right )^{-n_{1}}d x}{b_{1}}}\]

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