6.9.9 4.3

6.9.9.1 [1981] Problem 1

problem number 1981

Added Jan 19, 2020.

Problem Chapter 9.4.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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6.9.9.2 [1982] Problem 2

problem number 1982

Added Jan 19, 2020.

Problem Chapter 9.4.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y , z\right ) = \frac {{\mathrm e}^{\frac {q x}{a}} \cosh \left (\lambda x \right )^{\frac {p}{a \lambda }} \left (k \int \tanh \left (\gamma x \right ) {\mathrm e}^{-\frac {q x}{a}} \cosh \left (\lambda x \right )^{-\frac {p}{a \lambda }}d x +f_{1} \left (y -\frac {b x}{a}, -x +\frac {\ln \left (-\frac {\tanh \left (\beta z \right )}{\sqrt {-\operatorname {sech}\left (\beta z \right )^{2}}}\right ) a}{c \beta }\right ) a \right )}{a}\]

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6.9.9.3 [1983] Problem 3

problem number 1983

Added Jan 19, 2020.

Problem Chapter 9.4.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to -\frac {k \left (e^{-2 \mu x}-1\right )^m \left (e^{-2 \mu x}+1\right )^m \left (-e^{-4 \mu x} \left (e^{2 \mu x}-1\right )^2\right )^{-m} \tanh ^m(\mu x) \operatorname {AppellF1}\left (\frac {c}{2 \mu },m,-m,\frac {c}{2 \mu }+1,-e^{-2 \mu x},e^{-2 \mu x}\right )}{c}+e^{c x} c_1\left (y-\frac {a \tanh ^{n+1}(\beta x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\tanh ^2(\beta x)\right )}{\beta n+\beta },z-\frac {b \tanh ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {k+1}{2},\frac {k+3}{2},\tanh ^2(\lambda x)\right )}{k \lambda +\lambda }\right )\right \}\right \}\]

Maple ✓

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

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

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6.9.9.4 [1984] Problem 4

problem number 1984

Added Jan 19, 2020.

Problem Chapter 9.4.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to -\frac {s \left (e^{-2 \mu x}-1\right )^m \left (e^{-2 \mu x}+1\right )^m \left (-e^{-4 \mu x} \left (e^{2 \mu x}-1\right )^2\right )^{-m} \tanh ^m(\mu x) \operatorname {AppellF1}\left (\frac {a}{2 \mu },m,-m,\frac {a}{2 \mu }+1,-e^{-2 \mu x},e^{-2 \mu x}\right )}{a}+e^{a x} c_1\left (y-\frac {b \tanh ^{n+1}(\beta x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\tanh ^2(\beta x)\right )}{\beta n+\beta },z-\int _1^xc \tanh ^k\left (\frac {\lambda \left (-b \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\tanh ^2(\beta x)\right ) \tanh ^{n+1}(\beta x)+b \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\tanh ^2(\beta K[1])\right ) \tanh ^{n+1}(\beta K[1])+\beta (n+1) y\right )}{\beta (n+1)}\right )dK[1]\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde := diff(w(x,y,z),x)+ b*tanh(beta*x)^n*diff(w(x,y,z),y)+ c*tanh(lambda*y)^k*diff(w(x,y,z),z)=a*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'));
 

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

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6.9.9.5 [1985] Problem 5

problem number 1985

Added Jan 19, 2020.

Problem Chapter 9.4.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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