7.9.7 4.1

7.9.7.1 [1973] Problem 1
7.9.7.2 [1974] Problem 2
7.9.7.3 [1975] Problem 3
7.9.7.4 [1976] Problem 4
7.9.7.5 [1977] Problem 5

7.9.7.1 [1973] Problem 1

problem number 1973

Added Jan 19, 2020.

Problem Chapter 9.4.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {c \sqrt {\cosh ^2(\beta x)} \text {sech}(\beta x) \sinh ^{n+1}(\beta x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};-\sinh ^2(\beta x)\right )}{\beta n+\beta }\right ) \left (\int _1^x\exp \left (-\frac {c \sqrt {\cosh ^2(\beta K[1])} \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};-\sinh ^2(\beta K[1])\right ) \text {sech}(\beta K[1]) \sinh ^{n+1}(\beta K[1])}{n \beta +\beta }\right ) k \sinh ^m(\lambda K[1])dK[1]+c_1(y-a x,z-b x)\right )\right \}\right \}\]

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

\[w \left (x , y , z\right ) = \left (\int k \left (\sinh ^{m}\left (\lambda x \right )\right ) {\mathrm e}^{-c \left (\int \left (\sinh ^{n}\left (\beta x \right )\right )d x \right )}d x +\textit {\_F1} \left (-a x +y , -b x +z \right )\right ) {\mathrm e}^{\int c \left (\sinh ^{n}\left (\beta x \right )\right )d x}\]

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7.9.7.2 [1974] Problem 2

problem number 1974

Added Jan 19, 2020.

Problem Chapter 9.4.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica

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

\[\left \{\left \{w(x,y,z)\to e^{\frac {p \cosh (\lambda x)+\lambda q x}{a \lambda }} \left (\int _1^x\frac {e^{-\frac {p \cosh (\lambda K[1])+\lambda q K[1]}{a \lambda }} k \sinh (\gamma K[1])}{a}dK[1]+c_1\left (y-\frac {b x}{a},\frac {\log \left (\tanh \left (\frac {\beta z}{2}\right )\right )}{\beta }-\frac {c x}{a}\right )\right )\right \}\right \}\]

Maple

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

\[w \left (x , y , z\right ) = \left (\int \frac {k \,{\mathrm e}^{\frac {-\lambda q x -p \cosh \left (\lambda x \right )}{a \lambda }} \sinh \left (\gamma x \right )}{a}d x +\textit {\_F1} \left (\frac {a y -b x}{a}, \frac {-\beta c x -2 a \arctanh \left ({\mathrm e}^{\beta z}\right )}{\beta c}\right )\right ) {\mathrm e}^{\frac {\lambda q x +p \cosh \left (\lambda x \right )}{a \lambda }}\]

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7.9.7.3 [1975] Problem 3

problem number 1975

Added Jan 19, 2020.

Problem Chapter 9.4.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica

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

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

Maple

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

\[w \left (x , y , z\right ) = \left (\int k \left (\sinh ^{m}\left (\mu x \right )\right ) {\mathrm e}^{-c x}d x +\textit {\_F1} \left (y -\left (\int a \left (\sinh ^{n}\left (\beta x \right )\right )d x \right ), z -\left (\int b \left (\sinh ^{k}\left (\lambda x \right )\right )d x \right )\right )\right ) {\mathrm e}^{c x}\]

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7.9.7.4 [1976] Problem 4

problem number 1976

Added Jan 19, 2020.

Problem Chapter 9.4.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica

ClearAll["Global`*"]; 
pde =  D[w[x,y,z],x]+ b*Sinh[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*Sinh[mu*x]^m; 
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)+ b*sinh(beta*x)^n*diff(w(x,y,z),y)+ c*sinh(lambda*y)^k*diff(w(x,y,z),z)=a*w(x,y,z)+ s*sinh(mu*x)^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \left (\int s \left (\sinh ^{m}\left (\mu x \right )\right ) {\mathrm e}^{-a x}d x +\textit {\_F1} \left (y -\left (\int b \left (\sinh ^{n}\left (\beta x \right )\right )d x \right ), z -\left (\int _{}^{x}c \left (\sinh ^{k}\left (\left (b \left (\int \left (\sinh ^{n}\left (\textit {\_b} \beta \right )\right )d \textit {\_b} \right )+y -\left (\int b \left (\sinh ^{n}\left (\beta x \right )\right )d x \right )\right ) \lambda \right )\right )d \textit {\_b} \right )\right )\right ) {\mathrm e}^{a x}\]

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7.9.7.5 [1977] Problem 5

problem number 1977

Added Jan 19, 2020.

Problem Chapter 9.4.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica

ClearAll["Global`*"]; 
pde =  b1*Sinh[lambda1*x]^n1*D[w[x,y,z],x]+ b2*Sinh[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*Sinh[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]];
 

\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {a \sqrt {\cosh ^2(\text {lambda1} x)} \text {sech}(\text {lambda1} x) \sinh ^{1-\text {n1}}(\text {lambda1} x) \, _2F_1\left (\frac {1}{2},\frac {1-\text {n1}}{2};\frac {3-\text {n1}}{2};-\sinh ^2(\text {lambda1} x)\right )}{\text {b1} \text {lambda1}-\text {b1} \text {lambda1} \text {n1}}\right ) \left (\int _1^x\frac {\exp \left (\frac {a \sqrt {\cosh ^2(\text {lambda1} K[3])} \, _2F_1\left (\frac {1}{2},\frac {1-\text {n1}}{2};\frac {3-\text {n1}}{2};-\sinh ^2(\text {lambda1} K[3])\right ) \text {sech}(\text {lambda1} K[3]) \sinh ^{1-\text {n1}}(\text {lambda1} K[3])}{\text {b1} \text {lambda1} \text {n1}-\text {b1} \text {lambda1}}\right ) \left (\text {c1} \sinh ^{\text {k1}}(\text {beta1} K[3])+\text {c2} \sinh ^{\text {k2}}(\text {beta2} K[3])+\text {c3} \sinh ^{\text {k3}}(\text {beta3} K[3])\right ) \sinh ^{-\text {n1}}(\text {lambda1} K[3])}{\text {b1}}dK[3]+c_1\left (y-\int _1^x\frac {\text {b2} \sinh ^{-\text {n1}}(\text {lambda1} K[1]) \sinh ^{\text {n2}}(\text {lambda2} K[1])}{\text {b1}}dK[1],z-\int _1^x\frac {\text {b3} \sinh ^{-\text {n1}}(\text {lambda1} K[2]) \sinh ^{\text {n3}}(\text {lambda3} K[2])}{\text {b1}}dK[2]\right )\right )\right \}\right \}\]

Maple

restart; 
local gamma; 
pde := b__1*sinh(lambda__1*x)^(n__1)*diff(w(x,y,z),x)+b__2*sinh(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*sinh(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'));
 

\[w \left (x , y , z\right ) = \left (\int \frac {\left (c_{1} \left (\sinh ^{k_{1}}\left (\beta _{1} x \right )\right )+c_{2} \left (\sinh ^{k_{2}}\left (\beta _{2} x \right )\right )+c_{3} \left (\sinh ^{k_{3}}\left (\beta _{3} x \right )\right )\right ) \left (\sinh ^{-n_{1}}\left (x \lambda _{1} \right )\right ) {\mathrm e}^{-\frac {a \left (\int \left (\sinh ^{-n_{1}}\left (x \lambda _{1} \right )\right )d x \right )}{b_{1}}}}{b_{1}}d x +\textit {\_F1} \left (\frac {b_{1} y -b_{2} \left (\int \left (\sinh ^{-n_{1}}\left (x \lambda _{1} \right )\right ) \left (\sinh ^{n_{2}}\left (x \lambda _{2} \right )\right )d x \right )}{b_{1}}, \frac {b_{1} z -b_{3} \left (\int \left (\sinh ^{-n_{1}}\left (x \lambda _{1} \right )\right ) \left (\sinh ^{n_{3}}\left (x \lambda _{3} \right )\right )d x \right )}{b_{1}}\right )\right ) {\mathrm e}^{\int \frac {a \left (\sinh ^{-n_{1}}\left (x \lambda _{1} \right )\right )}{b_{1}}d x}\]

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