6.8.10 4.4

6.8.10.1 [1821] Problem 1

problem number 1821

Added Oct 10, 2019.

Problem Chapter 8.4.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.8.10.2 [1822] Problem 2

problem number 1822

Added Oct 10, 2019.

Problem Chapter 8.4.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.8.10.3 [1823] Problem 3

problem number 1823

Added Oct 10, 2019.

Problem Chapter 8.4.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.8.10.4 [1824] Problem 4

problem number 1824

Added Oct 10, 2019.

Problem Chapter 8.4.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*Coth[beta*y]*D[w[x, y,z], y] +  c*Coth[lambda*x]*D[w[x,y,z],z]== k*Coth[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*coth(beta*y)*diff(w(x, y,z), y) +  c*coth(lambda*x)*diff(w(x,y,z),z)= k*coth(gamma*z) *w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.8.10.5 [1825] Problem 5

problem number 1825

Added Oct 10, 2019.

Problem Chapter 8.4.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.8.10.6 [1826] Problem 6

problem number 1826

Added Oct 10, 2019.

Problem Chapter 8.4.4.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________