6.7.9 4.3

6.7.9.1 [1645] Problem 1

problem number 1645

Added June 20, 2019.

Problem Chapter 7.4.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y + b w_z = c \tanh ^k(\lambda x)+s \]

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

\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x)+\frac {c \tanh ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {k+1}{2},\frac {k+3}{2},\tanh ^2(\lambda x)\right )}{k \lambda +\lambda }+s x\right \}\right \}\]

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

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

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6.7.9.2 [1646] Problem 2

problem number 1646

Added June 20, 2019.

Problem Chapter 7.4.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y + c \tanh (\lambda x) w_z = k \tanh (\beta y)+s \tanh (\gamma z) \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y , z\right ) = \frac {\int _{}^{x}\frac {\left (\left (\cosh \left (\frac {\beta b \textit {\_a}}{a}\right ) \sinh \left (\frac {\gamma c \ln \left (\cosh \left (\lambda \textit {\_a} \right )\right )}{a \lambda }\right ) s +\sinh \left (\frac {\beta b \textit {\_a}}{a}\right ) k \cosh \left (\frac {\gamma c \ln \left (\cosh \left (\lambda \textit {\_a} \right )\right )}{a \lambda }\right )\right ) \cosh \left (\frac {\beta \left (a y -b x \right )}{a}\right )+\sinh \left (\frac {\beta \left (a y -b x \right )}{a}\right ) \left (\cosh \left (\frac {\beta b \textit {\_a}}{a}\right ) \cosh \left (\frac {\gamma c \ln \left (\cosh \left (\lambda \textit {\_a} \right )\right )}{a \lambda }\right ) k +\sinh \left (\frac {\beta b \textit {\_a}}{a}\right ) \sinh \left (\frac {\gamma c \ln \left (\cosh \left (\lambda \textit {\_a} \right )\right )}{a \lambda }\right ) s \right )\right ) \cosh \left (\frac {\gamma \left (-z a \lambda +c \ln \left (\cosh \left (\lambda x \right )\right )\right )}{a \lambda }\right )-\sinh \left (\frac {\gamma \left (-z a \lambda +c \ln \left (\cosh \left (\lambda x \right )\right )\right )}{a \lambda }\right ) \left (\left (\cosh \left (\frac {\beta b \textit {\_a}}{a}\right ) \cosh \left (\frac {\gamma c \ln \left (\cosh \left (\lambda \textit {\_a} \right )\right )}{a \lambda }\right ) s +\sinh \left (\frac {\beta b \textit {\_a}}{a}\right ) \sinh \left (\frac {\gamma c \ln \left (\cosh \left (\lambda \textit {\_a} \right )\right )}{a \lambda }\right ) k \right ) \cosh \left (\frac {\beta \left (a y -b x \right )}{a}\right )+\sinh \left (\frac {\beta \left (a y -b x \right )}{a}\right ) \left (\cosh \left (\frac {\beta b \textit {\_a}}{a}\right ) \sinh \left (\frac {\gamma c \ln \left (\cosh \left (\lambda \textit {\_a} \right )\right )}{a \lambda }\right ) k +\sinh \left (\frac {\beta b \textit {\_a}}{a}\right ) s \cosh \left (\frac {\gamma c \ln \left (\cosh \left (\lambda \textit {\_a} \right )\right )}{a \lambda }\right )\right )\right )}{\left (\cosh \left (\frac {\beta \left (a y -b x \right )}{a}\right ) \cosh \left (\frac {\beta b \textit {\_a}}{a}\right )+\sinh \left (\frac {\beta \left (a y -b x \right )}{a}\right ) \sinh \left (\frac {\beta b \textit {\_a}}{a}\right )\right ) \left (\cosh \left (\frac {\gamma \left (-z a \lambda +c \ln \left (\cosh \left (\lambda x \right )\right )\right )}{a \lambda }\right ) \cosh \left (\frac {\gamma c \ln \left (\cosh \left (\lambda \textit {\_a} \right )\right )}{a \lambda }\right )-\sinh \left (\frac {\gamma \left (-z a \lambda +c \ln \left (\cosh \left (\lambda x \right )\right )\right )}{a \lambda }\right ) \sinh \left (\frac {\gamma c \ln \left (\cosh \left (\lambda \textit {\_a} \right )\right )}{a \lambda }\right )\right )}d \textit {\_a}}{a}+f_{1} \left (\frac {a y -b x}{a}, \frac {z a \lambda -c \ln \left (\cosh \left (\lambda x \right )\right )}{a \lambda }\right )\]

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6.7.9.3 [1647] Problem 3

problem number 1647

Added June 19, 2019.

Problem Chapter 7.4.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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6.7.9.4 [1648] Problem 4

problem number 1648

Added June 19, 2019.

Problem Chapter 7.4.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \tanh (\beta y) w_y + c \tanh (\lambda x) w_z = k \tanh (\gamma z) \]

Mathematica ✗

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

Failed

Maple ✓

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

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

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6.7.9.5 [1649] Problem 5

problem number 1649

Added June 19, 2019.

Problem Chapter 7.4.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \tanh (\beta y) w_y + c \tanh (\gamma z) w_z = k \tanh (\lambda x) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to \frac {k \log (\cosh (\lambda x))}{a \lambda }+c_1\left (\frac {1}{2} \left (\frac {\log (\sinh (\gamma z))}{\gamma }-\frac {c x}{a}\right ),\frac {\log (\sinh (\beta y))}{\beta }-\frac {b \log (\sinh (\gamma z))}{c \gamma }\right )\right \}\right \}\]

Maple ✓

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

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

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6.7.9.6 [1650] Problem 6

problem number 1650

Added June 19, 2019.

Problem Chapter 7.4.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a \tanh (\lambda x) w_x + b \tanh (\beta y) w_y + c \tanh (\gamma z) w_z = k \]

Mathematica ✗

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

Failed

Maple ✓

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

\[w \left (x , y , z\right ) = \frac {f_{1} \left (\frac {-b \beta \ln \left ({\mathrm e}^{2 \lambda x}-1\right )+\ln \left (i \sinh \left (\beta y \right )\right ) a \lambda +b \beta \left (\lambda x +\ln \left (2\right )\right )}{\lambda b \beta }, \frac {-c \gamma \ln \left ({\mathrm e}^{2 \lambda x}-1\right )+\ln \left (i \sinh \left (\gamma z \right )\right ) a \lambda +c \gamma \left (\lambda x +\ln \left (2\right )\right )}{\lambda c \gamma }\right ) a \lambda +\ln \left (\sinh \left (\lambda x \right )\right ) k}{a \lambda }\]

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6.7.9.7 [1651] Problem 7

problem number 1651

Added June 19, 2019.

Problem Chapter 7.4.3.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a_1 \tanh ^{n_1}(\lambda _1 x) w_x + b_1 \tanh ^{m_1}(\beta _1 y) w_y + c_1 \tanh ^{k_1}(\gamma _1 z) w_z = a_2 \tanh ^{n_2}(\lambda _2 x) + b_2 \tanh ^{m_2}(\beta _2 y) w_y + c_2 \tanh ^{k_2}(\gamma _2 z) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a1*Tanh[lambda1*x]^n1*D[w[x, y,z], x] + b1*Tanh[beta1*x]^m1*D[w[x, y,z], y] +  c1*Tanh[gamma1*x]^k1*D[w[x,y,z],z]== a2*Tanh[lambda1*x]^n2 + b2*Tanh[beta2*x]^m2 +  c2*Tanh[gamma2*x]^k2; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \int _1^x\frac {\tanh ^{-\text {n1}}(\text {lambda1} K[3]) \left (\text {c2} \tanh ^{\text {k2}}(\text {gamma2} K[3])+\text {b2} \tanh ^{\text {m2}}(\text {beta2} K[3])+\text {a2} \tanh ^{\text {n2}}(\text {lambda1} K[3])\right )}{\text {a1}}dK[3]+c_1\left (y-\int _1^x\frac {\text {b1} \tanh ^{\text {m1}}(\text {beta1} K[1]) \tanh ^{-\text {n1}}(\text {lambda1} K[1])}{\text {a1}}dK[1],z-\int _1^x\frac {\text {c1} \tanh ^{\text {k1}}(\text {gamma1} K[2]) \tanh ^{-\text {n1}}(\text {lambda1} K[2])}{\text {a1}}dK[2]\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde := a1*tanh(lambda1*x)^n1*diff(w(x,y,z),x)+ b1*tanh(beta1*x)^m1*diff(w(x,y,z),y)+ c1*tanh(gamma1*x)^k1*diff(w(x,y,z),z)=a2*tanh(lambda1*x)^n2 + b2*tanh(beta2*x)^m2 +  c2*tanh(gamma2*x)^k2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \frac {\int \left (\operatorname {a2} \tanh \left (\lambda \operatorname {1} x \right )^{\operatorname {n2}}+\operatorname {b2} \tanh \left (\beta \operatorname {2} x \right )^{\operatorname {m2}}+\operatorname {c2} \tanh \left (\gamma \operatorname {2} x \right )^{\operatorname {k2}}\right ) \tanh \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x}{\operatorname {a1}}+f_{1} \left (-\frac {\operatorname {b1} \int \tanh \left (\beta \operatorname {1} x \right )^{\operatorname {m1}} \tanh \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x}{\operatorname {a1}}+y , -\frac {\operatorname {c1} \int \tanh \left (\gamma \operatorname {1} x \right )^{\operatorname {k1}} \tanh \left (\lambda \operatorname {1} x \right )^{-\operatorname {n1}}d x}{\operatorname {a1}}+z \right )\]

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