7.8.10 4.4

7.8.10.1 [1823] Problem 1
7.8.10.2 [1824] Problem 2
7.8.10.3 [1825] Problem 3
7.8.10.4 [1826] Problem 4
7.8.10.5 [1827] Problem 5
7.8.10.6 [1828] Problem 6

7.8.10.1 [1823] Problem 1

problem number 1823

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]];
 

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

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'));
 

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

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7.8.10.2 [1824] Problem 2

problem number 1824

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'));
 

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

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7.8.10.3 [1825] Problem 3

problem number 1825

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]];
 

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

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'));
 

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

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7.8.10.4 [1826] Problem 4

problem number 1826

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'));
 

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

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7.8.10.5 [1827] Problem 5

problem number 1827

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]];
 

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

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'));
 

\[w \left (x , y , z\right ) = \left (\coth \left (\lambda x \right )-1\right )^{-\frac {k}{2 a \lambda }} \left (\coth \left (\lambda x \right )+1\right )^{-\frac {k}{2 a \lambda }} \textit {\_F1} \left (\frac {-2 b \beta x +a \ln \left (\frac {\left (\RootOf \left (\beta y -\mathrm {arccoth}\left (\textit {\_Z} -1\right )\right )-1\right )^{2}}{\RootOf \left (\beta y -\mathrm {arccoth}\left (\textit {\_Z} -1\right )\right )-2}\right )-a \ln \left (\RootOf \left (\beta y -\mathrm {arccoth}\left (\textit {\_Z} -1\right )\right )\right )}{2 b \beta }, \frac {-2 c \gamma x +a \ln \left (\frac {\left (\RootOf \left (\gamma z -\mathrm {arccoth}\left (\textit {\_Z} -1\right )\right )-1\right )^{2}}{\RootOf \left (\gamma z -\mathrm {arccoth}\left (\textit {\_Z} -1\right )\right )-2}\right )-a \ln \left (\RootOf \left (\gamma z -\mathrm {arccoth}\left (\textit {\_Z} -1\right )\right )\right )}{2 c \gamma }\right )\]

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7.8.10.6 [1828] Problem 6

problem number 1828

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'));
 

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

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