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'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (-a x +y , -b x +z \right ) {\mathrm e}^{\int c \arctan \left (\beta x \right )^{n}d x}\]
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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]];
\[\left \{\left \{w(x,y,z)\to \left (\text {lambda1}^2 x^2+1\right )^{-\frac {\text {b1}}{2 \text {a1} \text {lambda1}}} \left (\text {a1}^2 \left (\text {lambda2}^2 y^2+1\right )\right )^{-\frac {\text {b2}}{2 \text {a2} \text {lambda2}}} \left (\text {a1}^2 \left (\text {lambda3}^2 z^2+1\right )\right )^{-\frac {\text {b3}}{2 \text {a3} \text {lambda3}}} c_1\left (y-\frac {\text {a2} x}{\text {a1}},z-\frac {\text {a3} x}{\text {a1}}\right ) \exp \left (\frac {\text {b1} x \tan ^{-1}(\text {lambda1} x)}{\text {a1}}+\frac {\text {b2} y \tan ^{-1}(\text {lambda2} y)}{\text {a2}}+\frac {\text {b3} z \tan ^{-1}(\text {lambda3} z)}{\text {a3}}\right )\right \}\right \}\]
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'));
\[w \left (x , y , z\right ) = \left (\lambda _{1}^{2} x^{2}+1\right )^{-\frac {b_{1}}{2 a_{1} \lambda _{1}}} \left (y^{2} \lambda _{2}^{2}+1\right )^{-\frac {b_{2}}{2 a_{2} \lambda _{2}}} \left (z^{2} \lambda _{3}^{2}+1\right )^{-\frac {b_{3}}{2 a_{3} \lambda _{3}}} \mathit {\_F1} \left (\frac {y a_{1} -x a_{2}}{a_{1}}, \frac {z a_{1} -a_{3} x}{a_{1}}\right ) {\mathrm e}^{\frac {a_{1} a_{3} b_{2} y \arctan \left (y \lambda _{2} \right )+\left (a_{1} b_{3} z \arctan \left (z \lambda _{3} \right )+a_{3} b_{1} x \arctan \left (x \lambda _{1} \right )\right ) a_{2}}{a_{1} a_{2} a_{3}}}\]
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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'));
\[w \left (x , y , z\right ) = \left (\gamma ^{2} x^{2}+1\right )^{-\frac {s}{2 a \gamma }} \mathit {\_F1} \left (\frac {a y -b x}{a}, -\left (\int \arctan \left (\lambda x \right )^{n}d x \right )+\int \frac {a \arctan \left (\beta z \right )^{-k}}{c}d z \right ) {\mathrm e}^{\frac {s x \arctan \left (\gamma x \right )}{a}}\]
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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'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {a y -b x}{a}, -\left (\int _{}^{x}\arctan \left (\lambda \mathit {\_a} \right )^{n} \arctan \left (\frac {\beta \left (\mathit {\_a} b +a y -b x \right )}{a}\right )^{m}d\mathit {\_a} \right )+\int \frac {\arctan \left (\gamma z \right )^{-k} a}{c}d z \right ) {\mathrm e}^{\frac {s x}{a}}\]
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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'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (-y +\int \frac {b \arctan \left (\lambda x \right )^{n}}{a}d x , \int \frac {b \arctan \left (\beta z \right )^{-k}}{c}d z -\left (\int _{}^{y}\arctan \left (\lambda \RootOf \left (\mathit {\_b} -y +\int \frac {b \arctan \left (\lambda x \right )^{n}}{a}d x -\left (\int _{}^{\mathit {\_Z}}\frac {b \arctan \left (\mathit {\_b} \lambda \right )^{n}}{a}d\mathit {\_b} \right )\right )\right )^{-n}d\mathit {\_b} \right )\right ) {\mathrm e}^{\int _{}^{y}\frac {s \arctan \left (\gamma \RootOf \left (\mathit {\_b} -y +\int \frac {b \arctan \left (\lambda x \right )^{n}}{a}d x -\left (\int _{}^{\mathit {\_Z}}\frac {b \arctan \left (\mathit {\_a} \lambda \right )^{n}}{a}d\mathit {\_a} \right )\right )\right )^{m} \arctan \left (\lambda \RootOf \left (\mathit {\_b} -y +\int \frac {b \arctan \left (\lambda x \right )^{n}}{a}d x -\left (\int _{}^{\mathit {\_Z}}\frac {b \arctan \left (\mathit {\_a} \lambda \right )^{n}}{a}d\mathit {\_a} \right )\right )\right )^{-n}}{b}d\mathit {\_b}}\]
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