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]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x)+\frac {c \coth ^{k+1}(\lambda x) \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};\coth ^2(\lambda x)\right )}{k \lambda +\lambda }+s x\right \}\right \}\]
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'));
\[w \left ( x,y,z \right ) =\int \!c \left ( {\rm coth} \left (x\lambda \right ) \right ) ^{k}\,{\rm d}x+sx+{\it \_F1} \left ( -ax+y,-xb+z \right ) \]
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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 ) ={\it \_F1} \left ( {\frac {ay-xb}{a}},{\frac {2\,za\lambda +c\ln \left ( {\rm coth} \left (x\lambda \right )-1 \right ) +c\ln \left ( {\rm coth} \left (x\lambda \right )+1 \right ) }{2\,a\lambda }} \right ) +\int ^{x}\!{\frac {1}{a} \left ( \left ( -k+s \right ) \sinh \left ( {\frac {\gamma \,c\ln \left ( {\rm coth} \left ({\it \_a}\,\lambda \right )-1 \right ) +\gamma \,c\ln \left ( {\rm coth} \left ({\it \_a}\,\lambda \right )+1 \right ) -\gamma \,c\ln \left ( {\rm coth} \left (x\lambda \right )-1 \right ) -\gamma \,c\ln \left ( {\rm coth} \left (x\lambda \right )+1 \right ) +2\, \left ( -az\gamma +\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) \right ) \lambda }{2\,a\lambda }} \right ) -\sinh \left ( {\frac {\gamma \,c\ln \left ( {\rm coth} \left ({\it \_a}\,\lambda \right )-1 \right ) +\gamma \,c\ln \left ( {\rm coth} \left ({\it \_a}\,\lambda \right )+1 \right ) -\gamma \,c\ln \left ( {\rm coth} \left (x\lambda \right )-1 \right ) -\gamma \,c\ln \left ( {\rm coth} \left (x\lambda \right )+1 \right ) -2\, \left ( az\gamma +\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) \right ) \lambda }{2\,a\lambda }} \right ) \left ( k+s \right ) \right ) \left ( \cosh \left ( {\frac {\gamma \,c\ln \left ( {\rm coth} \left ({\it \_a}\,\lambda \right )-1 \right ) +\gamma \,c\ln \left ( {\rm coth} \left ({\it \_a}\,\lambda \right )+1 \right ) -\gamma \,c\ln \left ( {\rm coth} \left (x\lambda \right )-1 \right ) -\gamma \,c\ln \left ( {\rm coth} \left (x\lambda \right )+1 \right ) -2\, \left ( az\gamma +\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) \right ) \lambda }{2\,a\lambda }} \right ) -\cosh \left ( {\frac {\gamma \,c\ln \left ( {\rm coth} \left ({\it \_a}\,\lambda \right )-1 \right ) +\gamma \,c\ln \left ( {\rm coth} \left ({\it \_a}\,\lambda \right )+1 \right ) -\gamma \,c\ln \left ( {\rm coth} \left (x\lambda \right )-1 \right ) -\gamma \,c\ln \left ( {\rm coth} \left (x\lambda \right )+1 \right ) +2\, \left ( -az\gamma +\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) \right ) \lambda }{2\,a\lambda }} \right ) \right ) ^{-1}}{d{\it \_a}}\]
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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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (z-\frac {b \coth ^{k+1}(\lambda x) \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};\coth ^2(\lambda x)\right )}{k \lambda +\lambda },y-\frac {a \coth ^{n+1}(\beta x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\coth ^2(\beta x)\right )}{\beta n+\beta }\right )+\frac {c \coth ^{m+1}(\gamma x) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};\coth ^2(\gamma x)\right )}{\gamma m+\gamma }+s x\right \}\right \}\]
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'));
\[w \left ( x,y,z \right ) =\int \!c \left ( {\rm coth} \left (x\gamma \right ) \right ) ^{m}\,{\rm d}x+sx+{\it \_F1} \left ( -\int \!a \left ( {\rm coth} \left (\beta \,x\right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( {\rm coth} \left (x\lambda \right ) \right ) ^{k}\,{\rm d}x+z \right ) \]
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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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {1}{2\,\beta \,b} \left ( -2\,b\beta \,x+a\ln \left ( {\frac { \left ( \RootOf \left ( \beta \,y-{\rm arccoth} \left ({\it \_Z}-1\right ) \right ) -1 \right ) ^{2}}{\RootOf \left ( \beta \,y-{\rm arccoth} \left ({\it \_Z}-1\right ) \right ) -2}} \right ) -\ln \left ( \RootOf \left ( \beta \,y-{\rm arccoth} \left ({\it \_Z}-1\right ) \right ) \right ) a \right ) },{\frac {2\,za\lambda +c\ln \left ( {\rm coth} \left (x\lambda \right )-1 \right ) +c\ln \left ( {\rm coth} \left (x\lambda \right )+1 \right ) }{2\,a\lambda }} \right ) -\int ^{x}\!{\frac {k}{a}\cosh \left ( {\frac {\gamma \, \left ( -2\,za\lambda -c\ln \left ( {\rm coth} \left (x\lambda \right )-1 \right ) -c\ln \left ( {\rm coth} \left (x\lambda \right )+1 \right ) +\ln \left ( {\rm coth} \left ({\it \_a}\,\lambda \right )-1 \right ) c+\ln \left ( {\rm coth} \left ({\it \_a}\,\lambda \right )+1 \right ) c \right ) }{2\,a\lambda }} \right ) \left ( \sinh \left ( {\frac {\gamma \, \left ( -2\,za\lambda -c\ln \left ( {\rm coth} \left (x\lambda \right )-1 \right ) -c\ln \left ( {\rm coth} \left (x\lambda \right )+1 \right ) +\ln \left ( {\rm coth} \left ({\it \_a}\,\lambda \right )-1 \right ) c+\ln \left ( {\rm coth} \left ({\it \_a}\,\lambda \right )+1 \right ) c \right ) }{2\,a\lambda }} \right ) \right ) ^{-1}}{d{\it \_a}}\]
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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'));
\[w \left ( x,y,z \right ) ={\frac {1}{2\,a\lambda } \left ( 2\,{\it \_F1} \left ( 1/2\,{\frac {1}{\beta \,b} \left ( -2\,b\beta \,x+a\ln \left ( {\frac { \left ( \RootOf \left ( \beta \,y-{\rm arccoth} \left ({\it \_Z}-1\right ) \right ) -1 \right ) ^{2}}{\RootOf \left ( \beta \,y-{\rm arccoth} \left ({\it \_Z}-1\right ) \right ) -2}} \right ) -\ln \left ( \RootOf \left ( \beta \,y-{\rm arccoth} \left ({\it \_Z}-1\right ) \right ) \right ) a \right ) },-1/2\,{\frac {1}{c\gamma } \left ( 2\,c\gamma \,x+\ln \left ( \RootOf \left ( \gamma \,z-{\rm arccoth} \left ({\it \_Z}-1\right ) \right ) \right ) a-a\ln \left ( {\frac { \left ( \RootOf \left ( \gamma \,z-{\rm arccoth} \left ({\it \_Z}-1\right ) \right ) -1 \right ) ^{2}}{\RootOf \left ( \gamma \,z-{\rm arccoth} \left ({\it \_Z}-1\right ) \right ) -2}} \right ) \right ) } \right ) a\lambda +2\,qx\lambda -p\ln \left ( {\rm coth} \left (x\lambda \right )-1 \right ) -p\ln \left ( {\rm coth} \left (x\lambda \right )+1 \right ) \right ) }\]
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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'));
\[w \left ( x,y,z \right ) =\int \!{\frac { \left ( {\rm coth} \left (\lambda 1\,x\right ) \right ) ^{-{\it n1}+{\it n2}}{\it a2}+ \left ( {\rm coth} \left (\lambda 1\,x\right ) \right ) ^{-{\it n1}}{\it b2}\, \left ( {\rm coth} \left (\beta 2\,x\right ) \right ) ^{{\it m2}}+ \left ( {\rm coth} \left (\lambda 1\,x\right ) \right ) ^{-{\it n1}}{\it c2}\, \left ( {\rm coth} \left (\gamma 2\,x\right ) \right ) ^{{\it k2}}}{{\it a1}}}\,{\rm d}x+{\it \_F1} \left ( {\frac {1}{{\it a1}} \left ( y{\it a1}-{\it b1}\,\int \! \left ( {\frac {\cosh \left ( \beta 1\,x \right ) }{\sinh \left ( \beta 1\,x \right ) }} \right ) ^{{\it m1}} \left ( {\frac {\cosh \left ( \lambda 1\,x \right ) }{\sinh \left ( \lambda 1\,x \right ) }} \right ) ^{-{\it n1}}\,{\rm d}x \right ) },{\frac {1}{{\it a1}} \left ( z{\it a1}-{\it c1}\,\int \! \left ( {\frac {\cosh \left ( \lambda 1\,x \right ) }{\sinh \left ( \lambda 1\,x \right ) }} \right ) ^{-{\it n1}} \left ( {\frac {\cosh \left ( \gamma 1\,x \right ) }{\sinh \left ( \gamma 1\,x \right ) }} \right ) ^{{\it k1}}\,{\rm d}x \right ) } \right ) \]
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