Added Oct 18, 2019.
Problem Chapter 8.6.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a w_y + b w_z = c \cot ^n(\beta x) w \]
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*Cot[beta*x]^n*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (-\frac {c \cot ^{n+1}(\beta x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\cot ^2(\beta x)\right )}{\beta n+\beta }\right ) c_1(y-a x,z-b x)\right \}\right \}\]
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*cot(beta*x)^n*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -ax+y,-bx+z \right ) {{\rm e}^{\int \!c \left ( \cot \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b w_y + c \cot (\beta z) w_z = \left ( k \cot (\lambda x)+s \cot (\gamma y) \right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] + c*Cot[beta*z]*D[w[x,y,z],z]== (k*Cot[lambda*x]+s*Cot[gamma*y])*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},\frac {\log (\sec (\beta z))}{\beta }-\frac {c x}{a}\right ) \exp \left (\frac {a \lambda s \log (\tan (\gamma y))+a \lambda s \log (\cos (\gamma y))+b \gamma k \log (\tan (\lambda x))+b \gamma k \log (\cos (\lambda x))}{a b \gamma \lambda }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+b*diff(w(x,y,z),y)+ c*cot(beta*z)*diff(w(x,y,z),z)= (k*cot(lambda*x)+s*cot(gamma*y))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {-ya+bx}{b}},{\frac {-2\,yc\beta +b\ln \left ( \left ( \cot \left ( \beta \,z \right ) \right ) ^{2}+1 \right ) -2\,b\ln \left ( \cot \left ( \beta \,z \right ) \right ) }{2\,c\beta }} \right ) \left ( \left ( \cot \left ( \lambda \,x \right ) \right ) ^{2}+1 \right ) ^{-{\frac {k}{2\,a\lambda }}} \left ( \left ( \cot \left ( \gamma \,y \right ) \right ) ^{2}+1 \right ) ^{-{\frac {s}{2\,\gamma \,b}}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a \cot ^n(\beta x) w_y + b \cot ^k(\lambda x) w_z = c \cot ^m(\gamma x) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Cot[beta*x]^n*D[w[x, y,z], y] + b*Cot[lambda*x]^k*D[w[x,y,z],z]== c*Cot[gamma*x]^m*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (-\frac {c \cot ^{m+1}(\gamma x) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\cot ^2(\gamma x)\right )}{\gamma m+\gamma }\right ) c_1\left (\frac {b \cot ^{k+1}(\lambda x) \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};-\cot ^2(\lambda x)\right )}{k \lambda +\lambda }+z,\frac {a \cot ^{n+1}(\beta x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\cot ^2(\beta x)\right )}{\beta n+\beta }+y\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+a*cot(beta*x)^n*diff(w(x,y,z),y)+ b*cot(lambda*x)^k*diff(w(x,y,z),z)= c*cot(gamma*x)^m*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!a \left ( \cot \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( \cot \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+z \right ) {{\rm e}^{\int \!c \left ( \cot \left ( \gamma \,x \right ) \right ) ^{m}\,{\rm d}x}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b \cot (\beta y) w_y + c \cot (\lambda x) w_z = k \cot (\gamma z) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Cot[beta*y]*D[w[x, y,z], y] + c*Cot[lambda*x]^m*D[w[x,y,z],z]== k*Cot[gamma*z]*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\begin {align*} & \left \{w(x,y,z)\to c_1\left (\frac {a \lambda m z+a \lambda z+c \cot ^{m+1}(\lambda x) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\cot ^2(\lambda x)\right )}{a \lambda m+a \lambda },\frac {\log (\sec (\beta y))}{\beta }-\frac {b x}{a}\right ) \exp \left (\int _1^x\frac {k \cot \left (\frac {\gamma \left (c \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\cot ^2(\lambda x)\right ) \cot ^{m+1}(\lambda x)+a \lambda (m+1) z-c \cot ^{m+1}(\lambda K[1]) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\cot ^2(\lambda K[1])\right )\right )}{a \lambda (m+1)}\right )}{a}dK[1]\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (\frac {a \lambda m z+a \lambda z+c \cot ^{m+1}(\lambda x) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\cot ^2(\lambda x)\right )}{a \lambda m+a \lambda },\frac {\log (\sec (\beta y))}{\beta }-\frac {b x}{a}\right ) \exp \left (\int _1^x\frac {k \cot \left (\frac {\gamma \left (c \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\cot ^2(\lambda x)\right ) \cot ^{m+1}(\lambda x)+a \lambda (m+1) z-c \cot ^{m+1}(\lambda K[2]) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\cot ^2(\lambda K[2])\right )\right )}{a \lambda (m+1)}\right )}{a}dK[2]\right )\right \}\\ \end {align*}
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+b*cot(beta*y)*diff(w(x,y,z),y)+ c*cot(lambda*x)^m*diff(w(x,y,z),z)= k*cot(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 ) ={\it \_F1} \left ( -{\frac {-2\,xb\beta +a\ln \left ( \left ( \cot \left ( \beta \,y \right ) \right ) ^{2}+1 \right ) -2\,a\ln \left ( \cot \left ( \beta \,y \right ) \right ) }{2\,b\beta }},-\int ^{y}\!{\frac {c}{b\cot \left ( \beta \,{\it \_a} \right ) } \left ( { \left ( \left ( \cot \left ( {\frac {a\lambda \,\ln \left ( \cot \left ( \beta \,{\it \_a} \right ) \right ) }{b\beta }} \right ) \cot \left ( {\frac {\lambda }{b\beta } \left ( xb\beta +a\ln \left ( \cot \left ( \beta \,y \right ) \right ) -{\frac {a\ln \left ( \left ( \cot \left ( \beta \,y \right ) \right ) ^{2}+1 \right ) }{2}} \right ) } \right ) +1 \right ) \cot \left ( {\frac {a\lambda \,\ln \left ( \left ( \cot \left ( \beta \,{\it \_a} \right ) \right ) ^{2}+1 \right ) }{2\,b\beta }} \right ) +\cot \left ( {\frac {\lambda }{b\beta } \left ( xb\beta +a\ln \left ( \cot \left ( \beta \,y \right ) \right ) -{\frac {a\ln \left ( \left ( \cot \left ( \beta \,y \right ) \right ) ^{2}+1 \right ) }{2}} \right ) } \right ) -\cot \left ( {\frac {a\lambda \,\ln \left ( \cot \left ( \beta \,{\it \_a} \right ) \right ) }{b\beta }} \right ) \right ) \left ( \left ( \cot \left ( {\frac {a\lambda \,\ln \left ( \cot \left ( \beta \,{\it \_a} \right ) \right ) }{b\beta }} \right ) -\cot \left ( {\frac {\lambda }{b\beta } \left ( xb\beta +a\ln \left ( \cot \left ( \beta \,y \right ) \right ) -{\frac {a\ln \left ( \left ( \cot \left ( \beta \,y \right ) \right ) ^{2}+1 \right ) }{2}} \right ) } \right ) \right ) \cot \left ( {\frac {a\lambda \,\ln \left ( \left ( \cot \left ( \beta \,{\it \_a} \right ) \right ) ^{2}+1 \right ) }{2\,b\beta }} \right ) +\cot \left ( {\frac {a\lambda \,\ln \left ( \cot \left ( \beta \,{\it \_a} \right ) \right ) }{b\beta }} \right ) \cot \left ( {\frac {\lambda }{b\beta } \left ( xb\beta +a\ln \left ( \cot \left ( \beta \,y \right ) \right ) -{\frac {a\ln \left ( \left ( \cot \left ( \beta \,y \right ) \right ) ^{2}+1 \right ) }{2}} \right ) } \right ) +1 \right ) ^{-1}} \right ) ^{m}}{d{\it \_a}}+z \right ) {{\rm e}^{\int ^{y}\!{\frac {k}{b\cot \left ( \beta \,{\it \_b} \right ) }\cot \left ( \gamma \, \left ( \int \!{\frac {c}{b\cot \left ( \beta \,{\it \_b} \right ) } \left ( { \left ( \cot \left ( {\frac {\lambda }{b\beta } \left ( xb\beta +a\ln \left ( \cot \left ( \beta \,y \right ) \right ) -{\frac {a\ln \left ( \left ( \cot \left ( \beta \,y \right ) \right ) ^{2}+1 \right ) }{2}} \right ) } \right ) + \left ( \cot \left ( {\frac {a\lambda \,\ln \left ( \cot \left ( \beta \,{\it \_b} \right ) \right ) }{b\beta }} \right ) \cot \left ( {\frac {\lambda }{b\beta } \left ( xb\beta +a\ln \left ( \cot \left ( \beta \,y \right ) \right ) -{\frac {a\ln \left ( \left ( \cot \left ( \beta \,y \right ) \right ) ^{2}+1 \right ) }{2}} \right ) } \right ) +1 \right ) \cot \left ( {\frac {a\lambda \,\ln \left ( \left ( \cot \left ( \beta \,{\it \_b} \right ) \right ) ^{2}+1 \right ) }{2\,b\beta }} \right ) -\cot \left ( {\frac {a\lambda \,\ln \left ( \cot \left ( \beta \,{\it \_b} \right ) \right ) }{b\beta }} \right ) \right ) \left ( \left ( \cot \left ( {\frac {a\lambda \,\ln \left ( \cot \left ( \beta \,{\it \_b} \right ) \right ) }{b\beta }} \right ) -\cot \left ( {\frac {\lambda }{b\beta } \left ( xb\beta +a\ln \left ( \cot \left ( \beta \,y \right ) \right ) -{\frac {a\ln \left ( \left ( \cot \left ( \beta \,y \right ) \right ) ^{2}+1 \right ) }{2}} \right ) } \right ) \right ) \cot \left ( {\frac {a\lambda \,\ln \left ( \left ( \cot \left ( \beta \,{\it \_b} \right ) \right ) ^{2}+1 \right ) }{2\,b\beta }} \right ) +\cot \left ( {\frac {a\lambda \,\ln \left ( \cot \left ( \beta \,{\it \_b} \right ) \right ) }{b\beta }} \right ) \cot \left ( {\frac {\lambda }{b\beta } \left ( xb\beta +a\ln \left ( \cot \left ( \beta \,y \right ) \right ) -{\frac {a\ln \left ( \left ( \cot \left ( \beta \,y \right ) \right ) ^{2}+1 \right ) }{2}} \right ) } \right ) +1 \right ) ^{-1}} \right ) ^{m}}\,{\rm d}{\it \_b}-\int ^{y}\!{\frac {c}{b\cot \left ( \beta \,{\it \_a} \right ) } \left ( { \left ( \left ( \cot \left ( {\frac {a\lambda \,\ln \left ( \cot \left ( \beta \,{\it \_a} \right ) \right ) }{b\beta }} \right ) \cot \left ( {\frac {\lambda }{b\beta } \left ( xb\beta +a\ln \left ( \cot \left ( \beta \,y \right ) \right ) -{\frac {a\ln \left ( \left ( \cot \left ( \beta \,y \right ) \right ) ^{2}+1 \right ) }{2}} \right ) } \right ) +1 \right ) \cot \left ( {\frac {a\lambda \,\ln \left ( \left ( \cot \left ( \beta \,{\it \_a} \right ) \right ) ^{2}+1 \right ) }{2\,b\beta }} \right ) +\cot \left ( {\frac {\lambda }{b\beta } \left ( xb\beta +a\ln \left ( \cot \left ( \beta \,y \right ) \right ) -{\frac {a\ln \left ( \left ( \cot \left ( \beta \,y \right ) \right ) ^{2}+1 \right ) }{2}} \right ) } \right ) -\cot \left ( {\frac {a\lambda \,\ln \left ( \cot \left ( \beta \,{\it \_a} \right ) \right ) }{b\beta }} \right ) \right ) \left ( \left ( \cot \left ( {\frac {a\lambda \,\ln \left ( \cot \left ( \beta \,{\it \_a} \right ) \right ) }{b\beta }} \right ) -\cot \left ( {\frac {\lambda }{b\beta } \left ( xb\beta +a\ln \left ( \cot \left ( \beta \,y \right ) \right ) -{\frac {a\ln \left ( \left ( \cot \left ( \beta \,y \right ) \right ) ^{2}+1 \right ) }{2}} \right ) } \right ) \right ) \cot \left ( {\frac {a\lambda \,\ln \left ( \left ( \cot \left ( \beta \,{\it \_a} \right ) \right ) ^{2}+1 \right ) }{2\,b\beta }} \right ) +\cot \left ( {\frac {a\lambda \,\ln \left ( \cot \left ( \beta \,{\it \_a} \right ) \right ) }{b\beta }} \right ) \cot \left ( {\frac {\lambda }{b\beta } \left ( xb\beta +a\ln \left ( \cot \left ( \beta \,y \right ) \right ) -{\frac {a\ln \left ( \left ( \cot \left ( \beta \,y \right ) \right ) ^{2}+1 \right ) }{2}} \right ) } \right ) +1 \right ) ^{-1}} \right ) ^{m}}{d{\it \_a}}+z \right ) \right ) }{d{\it \_b}}}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a_1 \cot ^{n_1}(\lambda _1 x) w_x + b_1 \cot ^{m_1}(\beta _1 y) w_y + c_1 \cot ^{k_1}(\gamma _1 z) w_z = \left ( a_2 \cot ^{n_2}(\lambda _2 x) + b_2 \cot ^{m_2}(\beta _2 y) + c_2 \cot ^{k_2}(\gamma _2 z) \right ) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a1*Cot[lambda1*z]^n1*D[w[x, y,z], x] + b1*Cot[beta1*y]^m1*D[w[x, y,z], y] + c1*Cot[gamma1*z]^k1*D[w[x,y,z],z]== (a2*Cot[lambda2*z]^n2 + b2*Cot[beta2*y]^m2 + c2*Cot[gamma2*z]^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*cot(lambda1*z)^n1*diff(w(x,y,z),x)+ b1*cot(beta1*y)^m1*diff(w(x,y,z),y)+ c1*cot(gamma1*z)^k1*diff(w(x,y,z),z)= (a2*cot(lambda2*z)^n2 + b2*cot(beta2*y)^m2 + c2*cot(gamma2*z)^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 ) ={\it \_F1} \left ( -\int \! \left ( \cot \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac { \left ( \cot \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}\,{\rm d}z,-\int ^{y}\!{\frac {{\it a1}\, \left ( \cot \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}}{{\it b1}} \left ( \cot \left ( \lambda 1\,\RootOf \left ( \int \! \left ( \cot \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \cot \left ( \gamma 1\,{\it \_b} \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}{d{\it \_b}}-\int \! \left ( \cot \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac { \left ( \cot \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{{\it n1}}}{d{\it \_f}}+x \right ) {{\rm e}^{\int ^{y}\!{\frac { \left ( \cot \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}}{{\it b1}} \left ( {\it a2}\, \left ( \cot \left ( \lambda 2\,\RootOf \left ( \int \! \left ( \cot \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \cot \left ( \gamma 1\,{\it \_a} \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}{d{\it \_a}}-\int \! \left ( \cot \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac { \left ( \cot \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{{\it n2}}+{\it c2}\, \left ( \cot \left ( \gamma 2\,\RootOf \left ( \int \! \left ( \cot \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \cot \left ( \gamma 1\,{\it \_a} \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}{d{\it \_a}}-\int \! \left ( \cot \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac { \left ( \cot \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{{\it k2}}+{\it b2}\, \left ( \cot \left ( \beta 2\,{\it \_f} \right ) \right ) ^{{\it m2}} \right ) }{d{\it \_f}}}}\]
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