Added June 20, 2019.
Problem Chapter 7.4.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a w_y + b w_z = c \coth ^k(\lambda x)+s \]
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*Coth[lambda*x]^k+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x)+\frac {c \coth ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {k+1}{2},\frac {k+3}{2},\coth ^2(\lambda x)\right )}{k \lambda +\lambda }+s x\right \}\right \}\]
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*coth(lambda*x)^k+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = s x +\int c \left (\coth ^{k}\left (\lambda x \right )\right )d x +\mathit {\_F1} \left (-a x +y , -b x +z \right )\]
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Added June 20, 2019.
Problem Chapter 7.4.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \coth (\lambda x) w_z = k \coth (\beta y)+s \coth (\gamma z) \]
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*y]+s*Coth[gamma*z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\frac {k \coth \left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )+s \coth \left (\frac {\gamma (a \lambda z-c \log (\sinh (\lambda x))+c \log (\sinh (\lambda K[1])))}{a \lambda }\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a},z-\frac {c \log (\sinh (\lambda x))}{a \lambda }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*coth(lambda*x)*diff(w(x,y,z),z)=k*coth(beta*y)+s*coth(gamma*z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{x}\frac {-\left (k +s \right ) \sinh \left (\frac {c \gamma \ln \left (\coth \left (\mathit {\_a} \lambda \right )-1\right )+c \gamma \ln \left (\coth \left (\mathit {\_a} \lambda \right )+1\right )-c \gamma \ln \left (\coth \left (\lambda x \right )-1\right )-c \gamma \ln \left (\coth \left (\lambda x \right )+1\right )-2 \left (a \gamma z +\left (a y -\left (-\mathit {\_a} +x \right ) b \right ) \beta \right ) \lambda }{2 a \lambda }\right )+\left (-k +s \right ) \sinh \left (\frac {c \gamma \ln \left (\coth \left (\mathit {\_a} \lambda \right )-1\right )+c \gamma \ln \left (\coth \left (\mathit {\_a} \lambda \right )+1\right )-c \gamma \ln \left (\coth \left (\lambda x \right )-1\right )-c \gamma \ln \left (\coth \left (\lambda x \right )+1\right )+2 \left (-a \gamma z +\left (a y -\left (-\mathit {\_a} +x \right ) b \right ) \beta \right ) \lambda }{2 a \lambda }\right )}{\left (\cosh \left (\frac {c \gamma \ln \left (\coth \left (\mathit {\_a} \lambda \right )-1\right )+c \gamma \ln \left (\coth \left (\mathit {\_a} \lambda \right )+1\right )-c \gamma \ln \left (\coth \left (\lambda x \right )-1\right )-c \gamma \ln \left (\coth \left (\lambda x \right )+1\right )-2 \left (a \gamma z +\left (a y -\left (-\mathit {\_a} +x \right ) b \right ) \beta \right ) \lambda }{2 a \lambda }\right )-\cosh \left (\frac {c \gamma \ln \left (\coth \left (\mathit {\_a} \lambda \right )-1\right )+c \gamma \ln \left (\coth \left (\mathit {\_a} \lambda \right )+1\right )-c \gamma \ln \left (\coth \left (\lambda x \right )-1\right )-c \gamma \ln \left (\coth \left (\lambda x \right )+1\right )+2 \left (-a \gamma z +\left (a y -\left (-\mathit {\_a} +x \right ) b \right ) \beta \right ) \lambda }{2 a \lambda }\right )\right ) a}d\mathit {\_a} +\mathit {\_F1} \left (\frac {a y -b x}{a}, \frac {2 a \lambda z +c \ln \left (\coth \left (\lambda x \right )-1\right )+c \ln \left (\coth \left (\lambda x \right )+1\right )}{2 a \lambda }\right )\]
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Added June 20, 2019.
Problem Chapter 7.4.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \coth ^n(\beta x) w_y + b \coth ^k(\lambda x) w_z = c \coth ^m(\gamma x)+s \]
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+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (z-\frac {b \coth ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {k+1}{2},\frac {k+3}{2},\coth ^2(\lambda x)\right )}{k \lambda +\lambda },y-\frac {a \coth ^{n+1}(\beta x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\coth ^2(\beta x)\right )}{\beta n+\beta }\right )+\frac {c \coth ^{m+1}(\gamma x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},\coth ^2(\gamma x)\right )}{\gamma m+\gamma }+s x\right \}\right \}\]
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+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = s x +\int c \left (\coth ^{m}\left (\gamma x \right )\right )d x +\mathit {\_F1} \left (y -\left (\int a \left (\coth ^{n}\left (\beta x \right )\right )d x \right ), z -\left (\int b \left (\coth ^{k}\left (\lambda x \right )\right )d x \right )\right )\]
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Added June 20, 2019.
Problem Chapter 7.4.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \coth (\beta y) w_y + c \coth (\lambda x) w_z = k \coth (\gamma z) \]
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]; 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); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = -\left (\int _{}^{x}\frac {k \cosh \left (\frac {\left (-2 a \lambda z +c \ln \left (\coth \left (\mathit {\_a} \lambda \right )-1\right )+c \ln \left (\coth \left (\mathit {\_a} \lambda \right )+1\right )-c \ln \left (\coth \left (\lambda x \right )-1\right )-c \ln \left (\coth \left (\lambda x \right )+1\right )\right ) \gamma }{2 a \lambda }\right )}{a \sinh \left (\frac {\left (-2 a \lambda z +c \ln \left (\coth \left (\mathit {\_a} \lambda \right )-1\right )+c \ln \left (\coth \left (\mathit {\_a} \lambda \right )+1\right )-c \ln \left (\coth \left (\lambda x \right )-1\right )-c \ln \left (\coth \left (\lambda x \right )+1\right )\right ) \gamma }{2 a \lambda }\right )}d\mathit {\_a} \right )+\mathit {\_F1} \left (\frac {-2 b \beta x +a \ln \left (\frac {\left (\RootOf \left (\beta y -\mathrm {arccoth}\left (\mathit {\_Z} -1\right )\right )-1\right )^{2}}{\RootOf \left (\beta y -\mathrm {arccoth}\left (\mathit {\_Z} -1\right )\right )-2}\right )-a \ln \left (\RootOf \left (\beta y -\mathrm {arccoth}\left (\mathit {\_Z} -1\right )\right )\right )}{2 b \beta }, \frac {2 a \lambda z +c \ln \left (\coth \left (\lambda x \right )-1\right )+c \ln \left (\coth \left (\lambda x \right )+1\right )}{2 a \lambda }\right )\]
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Added June 20, 2019.
Problem Chapter 7.4.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \coth (\beta y) w_y + c \coth (\gamma z) w_z = p \coth (\lambda x)+q \]
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]== p*Coth[lambda*x]+q; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \frac {a \lambda q \log (\cosh (\beta y))+b \beta p \log (-\tanh (\lambda x))+b \beta p \log (\cosh (\lambda x))}{a b \beta \lambda }+c_1\left (\frac {a \log (\text {sech}(\beta y))+b \beta x}{2 a \beta },\frac {2 c \log (\text {sech}(\beta y))}{\beta }-\frac {b \log \left (\text {sech}^2(\gamma z)\right )}{\gamma }\right )\right \}\right \}\]
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)=p*coth(lambda*x)+q; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {2 a \lambda \mathit {\_F1} \left (\frac {-2 b \beta x +a \ln \left (\frac {\left (\RootOf \left (\beta y -\mathrm {arccoth}\left (\mathit {\_Z} -1\right )\right )-1\right )^{2}}{\RootOf \left (\beta y -\mathrm {arccoth}\left (\mathit {\_Z} -1\right )\right )-2}\right )-a \ln \left (\RootOf \left (\beta y -\mathrm {arccoth}\left (\mathit {\_Z} -1\right )\right )\right )}{2 b \beta }, \frac {-2 c \gamma x +a \ln \left (\frac {\left (\RootOf \left (\gamma z -\mathrm {arccoth}\left (\mathit {\_Z} -1\right )\right )-1\right )^{2}}{\RootOf \left (\gamma z -\mathrm {arccoth}\left (\mathit {\_Z} -1\right )\right )-2}\right )-a \ln \left (\RootOf \left (\gamma z -\mathrm {arccoth}\left (\mathit {\_Z} -1\right )\right )\right )}{2 c \gamma }\right )+2 \lambda q x -p \ln \left (\coth \left (\lambda x \right )-1\right )-p \ln \left (\coth \left (\lambda x \right )+1\right )}{2 a \lambda }\]
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Added June 20, 2019.
Problem Chapter 7.4.4.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a_1 \coth ^{n_1}(\lambda _1 x) w_x + b_1 \coth ^{m_1}(\beta _1 y) w_y + c_1 \coth ^{k_1}(\gamma _1 z) w_z = a_2 \coth ^{n_2}(\lambda _2 x) + b_2 \coth ^{m_2}(\beta _2 y) w_y + c_2 \coth ^{k_2}(\gamma _2 z) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a1*Coth[lambda1*x]^n1*D[w[x, y,z], x] + b1*Coth[beta1*x]^m1*D[w[x, y,z], y] + c1*Coth[gamma1*x]^k1*D[w[x,y,z],z]== a2*Coth[lambda1*x]^n2 + b2*Coth[beta2*x]^m2 + c2*Coth[gamma2*x]^k2; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\frac {\coth ^{-\text {n1}}(\text {lambda1} K[3]) \left (\text {c2} \coth ^{\text {k2}}(\text {gamma2} K[3])+\text {b2} \coth ^{\text {m2}}(\text {beta2} K[3])+\text {a2} \coth ^{\text {n2}}(\text {lambda1} K[3])\right )}{\text {a1}}dK[3]+c_1\left (y-\int _1^x\frac {\text {b1} \coth ^{\text {m1}}(\text {beta1} K[1]) \coth ^{-\text {n1}}(\text {lambda1} K[1])}{\text {a1}}dK[1],z-\int _1^x\frac {\text {c1} \coth ^{\text {k1}}(\text {gamma1} K[2]) \coth ^{-\text {n1}}(\text {lambda1} K[2])}{\text {a1}}dK[2]\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a1*coth(lambda1*x)^n1*diff(w(x,y,z),x)+ b1*coth(beta1*x)^m1*diff(w(x,y,z),y)+ c1*coth(gamma1*x)^k1*diff(w(x,y,z),z)=a2*coth(lambda1*x)^n2 + b2*coth(beta2*x)^m2 + c2*coth(gamma2*x)^k2; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int \frac {\mathit {b2} \left (\coth ^{\mathit {m2}}\left (\beta 2 x \right )\right ) \left (\coth ^{-\mathit {n1}}\left (\lambda 1 x \right )\right )+\mathit {c2} \left (\coth ^{\mathit {k2}}\left (\gamma 2 x \right )\right ) \left (\coth ^{-\mathit {n1}}\left (\lambda 1 x \right )\right )+\mathit {a2} \left (\coth ^{-\mathit {n1} +\mathit {n2}}\left (\lambda 1 x \right )\right )}{\mathit {a1}}d x +\mathit {\_F1} \left (\frac {\mathit {a1} y -\mathit {b1} \left (\int \left (\frac {\cosh \left (\beta 1 x \right )}{\sinh \left (\beta 1 x \right )}\right )^{\mathit {m1}} \left (\frac {\cosh \left (\lambda 1 x \right )}{\sinh \left (\lambda 1 x \right )}\right )^{-\mathit {n1}}d x \right )}{\mathit {a1}}, \frac {\mathit {a1} z -\mathit {c1} \left (\int \left (\frac {\cosh \left (\gamma 1 x \right )}{\sinh \left (\gamma 1 x \right )}\right )^{\mathit {k1}} \left (\frac {\cosh \left (\lambda 1 x \right )}{\sinh \left (\lambda 1 x \right )}\right )^{-\mathit {n1}}d x \right )}{\mathit {a1}}\right )\]
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