Added June 26, 2019.
Problem Chapter 7.7.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a w_y + b w_z = c \arctan ^k(\lambda x)+s \]
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*ArcTan[lambda*x]^k+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^x\left (c \tan ^{-1}(\lambda K[1])^k+s\right )dK[1]+c_1(y-a x,z-b x)\right \}\right \}\]
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*arctan(lambda*x)^k+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = s x +\int c \arctan \left (\lambda x \right )^{k}d x +\textit {\_F1} \left (-a x +y , -b x +z \right )\]
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Added June 26, 2019.
Problem Chapter 7.7.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a_1 w_x + a_2 w_y + a_3 w_z = b_1 \arctan (\lambda _1 x)+b_2 \arctan (\lambda _2 y)+b_3 \arctan (\lambda _3 z) \]
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]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {\text {a2} x}{\text {a1}},z-\frac {\text {a3} x}{\text {a1}}\right )-\frac {\text {b2} \log \left (\text {a1}^2 \left (\text {lambda2}^2 y^2+1\right )\right )}{2 \text {a2} \text {lambda2}}-\frac {\text {b3} \log \left (\text {a1}^2 \left (\text {lambda3}^2 z^2+1\right )\right )}{2 \text {a3} \text {lambda3}}-\frac {\text {b1} \log \left (\text {lambda1}^2 x^2+1\right )}{2 \text {a1} \text {lambda1}}+\frac {\text {b1} x \tan ^{-1}(\text {lambda1} x)}{\text {a1}}+\frac {\text {b2} y \tan ^{-1}(\text {lambda2} y)}{\text {a2}}+\frac {\text {b3} z \tan ^{-1}(\text {lambda3} z)}{\text {a3}}\right \}\right \}\]
Maple ✓
restart; local gamma; pde := a1*diff(w(x,y,z),x)+ a2*diff(w(x,y,z),y)+ a3*diff(w(x,y,z),z)= b1*arctan(lambda1*x)+b2*arctan(lambda2*y)+b3*arctan(lambda3*z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {\mathit {b1} x \arctan \left (\lambda 1 x \right )}{\mathit {a1}}+\frac {\mathit {b2} y \arctan \left (\lambda 2 y \right )}{\mathit {a2}}+\frac {\mathit {b3} z \arctan \left (\lambda 3 z \right )}{\mathit {a3}}+\frac {2 \mathit {a1} \mathit {a2} \mathit {a3} \lambda 1 \lambda 2 \lambda 3 \textit {\_F1} \left (\frac {\mathit {a1} y -\mathit {a2} x}{\mathit {a1}}, \frac {\mathit {a1} z -\mathit {a3} x}{\mathit {a1}}\right )-\mathit {a2} \mathit {a3} \mathit {b1} \lambda 2 \lambda 3 \ln \left (\lambda 1^{2} x^{2}+1\right )-\left (\mathit {a2} \mathit {b3} \lambda 2 \ln \left (\lambda 3^{2} z^{2}+1\right )+\mathit {a3} \mathit {b2} \lambda 3 \ln \left (\lambda 2^{2} y^{2}+1\right )\right ) \mathit {a1} \lambda 1}{2 \mathit {a1} \mathit {a2} \mathit {a3} \lambda 1 \lambda 2 \lambda 3}\]
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Added June 26, 2019.
Problem Chapter 7.7.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \arctan ^n(\lambda x) \arctan ^k(\beta z) w_z = s \arctan ^m(\gamma x) \]
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; 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)^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int \frac {s \arctan \left (\gamma x \right )^{m}}{a}d x +\textit {\_F1} \left (\frac {a y -b x}{a}, -\left (\int \arctan \left (\lambda x \right )^{n}d x \right )+\int \frac {a \arctan \left (\beta z \right )^{-k}}{c}d z \right )\]
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Added June 26, 2019.
Problem Chapter 7.7.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \arctan ^n(\lambda x) \arctan ^m(\beta y) w_z = s \]
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*D[w[x,y,z],z]== s; 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*diff(w(x,y,z),z)= s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \frac {s x}{a}+\textit {\_F1} \left (\frac {a y -b x}{a}, z -\left (\int _{}^{x}\frac {c \arctan \left (\textit {\_a} \lambda \right )^{n} \arctan \left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \beta }{a}\right )^{m}}{a}d \textit {\_a} \right )\right )\]
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Added June 26, 2019.
Problem Chapter 7.7.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \arctan ^n(\lambda x) w_y + c \arctan ^k(\beta z) w_z = s \arctan ^m(\gamma x) \]
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; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \int _1^z\frac {s \tan ^{-1}(\beta K[3])^{-k} \tan ^{-1}\left (\frac {\gamma \left (c x-a \int _1^z\tan ^{-1}(\beta K[2])^{-k}dK[2]+a \int _1^{K[3]}\tan ^{-1}(\beta K[2])^{-k}dK[2]\right )}{c}\right ){}^m}{c}dK[3]+c_1\left (y-\int _1^x\frac {b \tan ^{-1}(\lambda K[1])^n}{a}dK[1],\int _1^z\tan ^{-1}(\beta K[2])^{-k}dK[2]-\frac {c x}{a}\right )\right \}\right \}\]
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; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \int _{}^{y}\frac {s \arctan \left (\gamma \RootOf \left (\textit {\_b} -y +\int \frac {b \arctan \left (\lambda x \right )^{n}}{a}d x -\left (\int _{}^{\textit {\_Z}}\frac {b \arctan \left (\textit {\_a} \lambda \right )^{n}}{a}d \textit {\_a} \right )\right )\right )^{m} \arctan \left (\lambda \RootOf \left (\textit {\_b} -y +\int \frac {b \arctan \left (\lambda x \right )^{n}}{a}d x -\left (\int _{}^{\textit {\_Z}}\frac {b \arctan \left (\textit {\_a} \lambda \right )^{n}}{a}d \textit {\_a} \right )\right )\right )^{-n}}{b}d \textit {\_b} +\textit {\_F1} \left (-y +\int \frac {b \arctan \left (\lambda x \right )^{n}}{a}d x , \int \frac {b \arctan \left (\beta z \right )^{-k}}{c}d z -\left (\int _{}^{y}\arctan \left (\lambda \RootOf \left (\textit {\_b} -y +\int \frac {b \arctan \left (\lambda x \right )^{n}}{a}d x -\left (\int _{}^{\textit {\_Z}}\frac {b \arctan \left (\textit {\_b} \lambda \right )^{n}}{a}d \textit {\_b} \right )\right )\right )^{-n}d \textit {\_b} \right )\right )\]
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