7.8.9 4.3

7.8.9.1 [1817] Problem 1

problem number 1817

Added Oct 10, 2019.

Problem Chapter 8.4.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {c \tanh ^{n+1}(\beta x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\tanh ^2(\beta x)\right )}{\beta n+\beta }\right ) c_1(y-a x,z-b x)\right \}\right \}\]

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

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

____________________________________________________________________________________

7.8.9.2 [1818] Problem 2

problem number 1818

Added Oct 10, 2019.

Problem Chapter 8.4.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {a y -b x}{a}, \frac {2 a \lambda z +c \ln \left (\tanh \left (\lambda x \right )-1\right )+c \ln \left (\tanh \left (\lambda x \right )+1\right )}{2 a \lambda }\right ) {\mathrm e}^{\int _{}^{x}\frac {k \tanh \left (\textit {\_a} \beta \right )-s \tanh \left (\frac {\left (-2 a \lambda z +c \ln \left (\tanh \left (\textit {\_a} \lambda \right )-1\right )+c \ln \left (\tanh \left (\textit {\_a} \lambda \right )+1\right )-c \ln \left (\tanh \left (\lambda x \right )-1\right )-c \ln \left (\tanh \left (\lambda x \right )+1\right )\right ) \gamma }{2 a \lambda }\right )}{a}d \textit {\_a}}\]

____________________________________________________________________________________

7.8.9.3 [1819] Problem 3

problem number 1819

Added Oct 10, 2019.

Problem Chapter 8.4.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {c \tanh ^{m+1}(\gamma x) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};\tanh ^2(\gamma x)\right )}{\gamma m+\gamma }\right ) c_1\left (y-\frac {a \tanh ^{n+1}(\beta x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\tanh ^2(\beta x)\right )}{\beta n+\beta },z-\frac {b \tanh ^{k+1}(\lambda x) \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};\tanh ^2(\lambda x)\right )}{k \lambda +\lambda }\right )\right \}\right \}\]

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

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

____________________________________________________________________________________

7.8.9.4 [1820] Problem 4

problem number 1820

Added Oct 10, 2019.

Problem Chapter 8.4.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {2 b \beta x +a \ln \left (\tanh \left (\beta y \right )-1\right )+a \ln \left (\tanh \left (\beta y \right )+1\right )-2 a \ln \left (\tanh \left (\beta y \right )\right )}{2 b \beta }, \frac {a \lambda z -c \ln \left (\cosh \left (\lambda x \right )\right )}{a \lambda }\right ) {\mathrm e}^{\int _{}^{y}\frac {k \tanh \left (\frac {\left (a \lambda z -c \ln \left (\cosh \left (\lambda x \right )\right )+c \ln \left (\cosh \left (\frac {\left (2 b \beta x -a \ln \left (\tanh \left (\textit {\_a} \beta \right )-1\right )-a \ln \left (\tanh \left (\textit {\_a} \beta \right )+1\right )+a \ln \left (\tanh \left (\beta y \right )-1\right )+a \ln \left (\tanh \left (\beta y \right )+1\right )+2 a \ln \left (\tanh \left (\textit {\_a} \beta \right )\right )-2 a \ln \left (\tanh \left (\beta y \right )\right )\right ) \lambda }{2 b \beta }\right )\right )\right ) \gamma }{a \lambda }\right )}{b \tanh \left (\textit {\_a} \beta \right )}d \textit {\_a}}\]

____________________________________________________________________________________

7.8.9.5 [1821] Problem 5

problem number 1821

Added Oct 10, 2019.

Problem Chapter 8.4.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

\[w \left (x , y , z\right ) = \left (\cosh ^{\frac {k}{a \lambda }}\left (\lambda x \right )\right ) \textit {\_F1} \left (\frac {2 b \beta x +a \ln \left (\tanh \left (\beta y \right )-1\right )+a \ln \left (\tanh \left (\beta y \right )+1\right )-2 a \ln \left (\tanh \left (\beta y \right )\right )}{2 b \beta }, \frac {b \beta \ln \left (\sqrt {-\left (-\frac {1}{\left ({\mathrm e}^{2 \beta y}-1\right )^{2}}\right )^{\frac {c \gamma }{b \beta }}}\, \sinh \left (\gamma z \right )\right )+\left (\beta y +\ln \left (2\right )\right ) c \gamma }{\beta c \gamma }\right )\]

____________________________________________________________________________________

7.8.9.6 [1822] Problem 6

problem number 1822

Added Oct 10, 2019.

Problem Chapter 8.4.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a1*Tanh[lambda1*x]^n1*D[w[x, y,z], x] + b1*Tanh[beta1*y]^m1*D[w[x, y,z], y] +  c1*Tanh[gamma1*x]^k1*D[w[x, y,z], z]== (a2*Tanh[lambda2*x]^n2+b2*Tanh[beta2*y]^m2+c2*Tanh[gamma2*x]^k2) *w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
local gamma; 
pde :=  a1*tanh(lambda1*x)^n1*diff(w(x, y,z), x) + b1*tanh(beta1*y)^m1*diff(w(x, y,z), y) +  c1*tanh(gamma1*x)^k1*diff(w(x,y,z),z)= ( a2*tanh(lambda2*x)^n2+b2*tanh(beta2*y)^m2+c2*tanh(gamma2*x)^k2) *w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \textit {\_F1} \left (-\left (\int \left (\tanh ^{-\mathit {n1}}\left (\lambda 1 x \right )\right )d x \right )+\int \frac {\mathit {a1} \left (\tanh ^{-\mathit {m1}}\left (\beta 1 y \right )\right )}{\mathit {b1}}d y , \frac {\mathit {a1} z -\mathit {c1} \left (\int \left (\frac {\sinh \left (\gamma 1 x \right )}{\cosh \left (\gamma 1 x \right )}\right )^{\mathit {k1}} \left (\frac {\sinh \left (\lambda 1 x \right )}{\cosh \left (\lambda 1 x \right )}\right )^{-\mathit {n1}}d x \right )}{\mathit {a1}}\right ) {\mathrm e}^{\int _{}^{x}\frac {\left (\mathit {a2} \left (\tanh ^{\mathit {n2}}\left (\textit {\_f} \lambda 2 \right )\right )+\mathit {b2} \left (\tanh ^{\mathit {m2}}\left (\beta 2 \RootOf \left (\int \left (\tanh ^{-\mathit {n1}}\left (\textit {\_f} \lambda 1 \right )\right )d \textit {\_f} -\left (\int \left (\tanh ^{-\mathit {n1}}\left (\lambda 1 x \right )\right )d x \right )+\int \frac {\mathit {a1} \left (\tanh ^{-\mathit {m1}}\left (\beta 1 y \right )\right )}{\mathit {b1}}d y -\left (\int ^{\textit {\_Z}}\frac {\mathit {a1} \left (\tanh ^{-\mathit {m1}}\left (\textit {\_a} \beta 1 \right )\right )}{\mathit {b1}}d \textit {\_a} \right )\right )\right )\right )+\mathit {c2} \left (\tanh ^{\mathit {k2}}\left (\textit {\_f} \gamma 2 \right )\right )\right ) \left (\tanh ^{-\mathit {n1}}\left (\textit {\_f} \lambda 1 \right )\right )}{\mathit {a1}}d \textit {\_f}}\]

____________________________________________________________________________________