Added Nov 30, 2019.
Problem Chapter 8.7.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a w_y + b w_z = c \arctan ^n(\beta x) w \]
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*ArcTan[beta*x]^n * w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x) \exp \left (\int _1^xc \tan ^{-1}(\beta K[1])^ndK[1]\right )\right \}\right \}\]
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*arctan(beta*x)^n*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -ax+y,-bx+z \right ) {{\rm e}^{\int \!c \left ( \arctan \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x}}\]
____________________________________________________________________________________
Added Nov 30, 2019.
Problem Chapter 8.7.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a_1 w_x + a_2 w_y + a_3 w_z = \left ( b_1 \arctan (\lambda _1 x)+b_2 \arctan (\lambda _2 y)+b_3 \arctan (\lambda _3 z) \right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a1*D[w[x,y,z],x]+a2*D[w[x,y,z],y]+a3*D[w[x,y,z],z]== (b1*ArcTan[lambda1*x]+b2*ArcTan[lambda2*y]+b3*ArcTan[lambda3*z] ) * w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \left (\text {lambda1}^2 x^2+1\right )^{-\frac {\text {b1}}{2 \text {a1} \text {lambda1}}} \left (\text {lambda2}^2 y^2+1\right )^{-\frac {\text {b2}}{2 \text {a2} \text {lambda2}}} \left (\text {lambda3}^2 z^2+1\right )^{-\frac {\text {b3}}{2 \text {a3} \text {lambda3}}} c_1\left (y-\frac {\text {a2} x}{\text {a1}},z-\frac {\text {a3} x}{\text {a1}}\right ) \exp \left (\frac {\text {b1} x \tan ^{-1}(\text {lambda1} x)}{\text {a1}}+\frac {\text {b2} y \tan ^{-1}(\text {lambda2} y)}{\text {a2}}+\frac {\text {b3} z \tan ^{-1}(\text {lambda3} z)}{\text {a3}}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a__1*diff(w(x,y,z),x)+ a__2*diff(w(x,y,z),y)+ a__3*diff(w(x,y,z),z)= (b__1*arctan(lambda__1*x)+b__2*arctan(lambda__2*y)+b__3*arctan(lambda__3*z))*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya_{1}-xa_{2}}{a_{1}}},{\frac {za_{1}-xa_{3}}{a_{1}}} \right ) \left ( {\lambda _{1}}^{2}{x}^{2}+1 \right ) ^{-{\frac {b_{1}}{2\,a_{1}\,\lambda _{1}}}} \left ( {y}^{2}{\lambda _{2}}^{2}+1 \right ) ^{-{\frac {b_{2}}{2\,a_{2}\,\lambda _{2}}}} \left ( {z}^{2}{\lambda _{3}}^{2}+1 \right ) ^{-{\frac {b_{3}}{2\,a_{3}\,\lambda _{3}}}}{{\rm e}^{{\frac {\arctan \left ( \lambda _{2}\,y \right ) ya_{1}\,a_{3}\,b_{2}+a_{2}\, \left ( x\arctan \left ( \lambda _{1}\,x \right ) a_{3}\,b_{1}+za_{1}\,b_{3}\,\arctan \left ( \lambda _{3}\,z \right ) \right ) }{a_{1}\,a_{2}\,a_{3}}}}}\]
____________________________________________________________________________________
Added Nov 30, 2019.
Problem Chapter 8.7.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b w_y + c \arctan ^n(\lambda x) \arctan ^k(\beta z) w_z = s \arctan ^m(\gamma x) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*D[w[x,y,z],x]+b*D[w[x,y,z],y]+c*ArcTan[lambda*x]^n*ArcTan[beta*z]^k*D[w[x,y,z],z]==s*ArcTan[gamma*x]^m * 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*diff(w(x,y,z),y)+ c*arctan(lambda*x)^n*arctan(beta*z)^k*diff(w(x,y,z),z)= s*arctan(gamma*x)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}},-\int \! \left ( \arctan \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+\int \!{\frac { \left ( \arctan \left ( \beta \,z \right ) \right ) ^{-k}a}{c}}\,{\rm d}z \right ) {{\rm e}^{{\frac {xs\arctan \left ( \gamma \,x \right ) }{a}}}} \left ( {\gamma }^{2}{x}^{2}+1 \right ) ^{-{\frac {s}{2\,a\gamma }}}\]
____________________________________________________________________________________
Added Nov 30, 2019.
Problem Chapter 8.7.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b w_y + c \arctan ^n(\lambda x) \arctan ^m(\beta y) \arctan ^k(\gamma z) w_z = s w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*D[w[x,y,z],x]+b*D[w[x,y,z],y]+c*ArcTan[lambda*x]^n*ArcTan[beta*y]^m*ArcTan[gamma*z]^k*D[w[x,y,z],z]==s* 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*diff(w(x,y,z),y)+ c*arctan(lambda*x)^n*arctan(beta*y)^m*arctan(gamma*z)^k*diff(w(x,y,z),z)= s*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}},-\int ^{x}\! \left ( \arctan \left ( \lambda \,{\it \_a} \right ) \right ) ^{n} \left ( \arctan \left ( {\frac {\beta \, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{m}{d{\it \_a}}+\int \!{\frac { \left ( \arctan \left ( \gamma \,z \right ) \right ) ^{-k}a}{c}}\,{\rm d}z \right ) {{\rm e}^{{\frac {sx}{a}}}}\]
____________________________________________________________________________________
Added Nov 30, 2019.
Problem Chapter 8.7.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a w_x + b \arctan ^n(\lambda x) w_y + c \arctan ^k(\beta z) w_z = s \arctan ^m(\gamma x) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x,y,z],x]+b*ArcTan[lambda*x]^n*D[w[x,y,z],y]+c*ArcTan[beta*z]^k*D[w[x,y,z],z]==s* ArcTan[gamma*x]^m*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\int _1^z\frac {s \tan ^{-1}(\beta K[3])^{-k} \tan ^{-1}\left (\frac {\gamma \left (c x-a \int _1^z\tan ^{-1}(\beta K[2])^{-k}dK[2]+a \int _1^{K[3]}\tan ^{-1}(\beta K[2])^{-k}dK[2]\right )}{c}\right ){}^m}{c}dK[3]\right ) c_1\left (y-\int _1^x\frac {b \tan ^{-1}(\lambda K[1])^n}{a}dK[1],\int _1^z\tan ^{-1}(\beta K[2])^{-k}dK[2]-\frac {c x}{a}\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a*diff(w(x,y,z),x)+ b*arctan(lambda*x)^n*diff(w(x,y,z),y)+ c*arctan(beta*z)^k*diff(w(x,y,z),z)= s*arctan(gamma*x)^m*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -y+\int \!{\frac {b \left ( \arctan \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x,-\int ^{y}\! \left ( \arctan \left ( \lambda \,\RootOf \left ( {\it \_b}-\int ^{{\it \_Z}}\!{\frac {b \left ( \arctan \left ( \lambda \,{\it \_b} \right ) \right ) ^{n}}{a}}{d{\it \_b}}-y+\int \!{\frac {b \left ( \arctan \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x \right ) \right ) \right ) ^{-n}{d{\it \_b}}+\int \!{\frac { \left ( \arctan \left ( \beta \,z \right ) \right ) ^{-k}b}{c}}\,{\rm d}z \right ) {{\rm e}^{\int ^{y}\!{\frac {s}{b} \left ( \arctan \left ( \gamma \,\RootOf \left ( {\it \_b}-\int ^{{\it \_Z}}\!{\frac {b \left ( \arctan \left ( \lambda \,{\it \_a} \right ) \right ) ^{n}}{a}}{d{\it \_a}}-y+\int \!{\frac {b \left ( \arctan \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x \right ) \right ) \right ) ^{m} \left ( \arctan \left ( \lambda \,\RootOf \left ( {\it \_b}-\int ^{{\it \_Z}}\!{\frac {b \left ( \arctan \left ( \lambda \,{\it \_a} \right ) \right ) ^{n}}{a}}{d{\it \_a}}-y+\int \!{\frac {b \left ( \arctan \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x \right ) \right ) \right ) ^{-n}}{d{\it \_b}}}}\]
____________________________________________________________________________________