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'));
\[w \left ( x,y,z \right ) =\int \!c \left ( \arctan \left ( x\lambda \right ) \right ) ^{k}\,{\rm d}x+sx+{\it \_F1} \left ( -ax+y,-xb+z \right ) \]
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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'));
\[w \left ( x,y,z \right ) ={\frac {{\it b1}\,\arctan \left ( \lambda 1\,x \right ) x}{{\it a1}}}+{\frac {{\it b2}\,\arctan \left ( \lambda 2\,y \right ) y}{{\it a2}}}+{\frac {{\it b3}\,\arctan \left ( \lambda 3\,z \right ) z}{{\it a3}}}+{\frac {1}{2\,\lambda 1\,\lambda 2\,{\it a2}\,{\it a1}\,\lambda 3\,{\it a3}} \left ( 2\,{\it \_F1} \left ( {\frac {y{\it a1}-{\it a2}\,x}{{\it a1}}},{\frac {z{\it a1}-{\it a3}\,x}{{\it a1}}} \right ) {\it a1}\,\lambda 1\,{\it a2}\,\lambda 2\,{\it a3}\,\lambda 3-{\it b1}\,\ln \left ( {\lambda 1}^{2}{x}^{2}+1 \right ) \lambda 2\,{\it a2}\,\lambda 3\,{\it a3}-{\it a1}\,\lambda 1\, \left ( {\it a2}\,{\it b3}\,\lambda 2\,\ln \left ( {\lambda 3}^{2}{z}^{2}+1 \right ) +{\it a3}\,{\it b2}\,\lambda 3\,\ln \left ( {\lambda 2}^{2}{y}^{2}+1 \right ) \right ) \right ) }\]
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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'));
\[w \left ( x,y,z \right ) =\int \!{\frac {s \left ( \arctan \left ( x\gamma \right ) \right ) ^{m}}{a}}\,{\rm d}x+{\it \_F1} \left ( {\frac {ay-xb}{a}},-\int \! \left ( \arctan \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+\int \!{\frac { \left ( \arctan \left ( \beta \,z \right ) \right ) ^{-k}a}{c}}\,{\rm d}z \right ) \]
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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'));
\[w \left ( x,y,z \right ) ={\frac {sx}{a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}},-\int ^{x}\!{\frac {c \left ( \arctan \left ( {\it \_a}\,\lambda \right ) \right ) ^{n}}{a} \left ( \arctan \left ( {\frac {\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{m}}{d{\it \_a}}+z \right ) \]
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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'));
\[w \left ( x,y,z \right ) =\int ^{y}\!{\frac {s}{b} \left ( \arctan \left ( \RootOf \left ( {\it \_b}-\int ^{{\it \_Z}}\!{\frac {b \left ( \arctan \left ( {\it \_a}\,\lambda \right ) \right ) ^{n}}{a}}{d{\it \_a}}-y+\int \!{\frac {b \left ( \arctan \left ( x\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}x \right ) \gamma \right ) \right ) ^{m} \left ( \arctan \left ( \RootOf \left ( {\it \_b}-\int ^{{\it \_Z}}\!{\frac {b \left ( \arctan \left ( {\it \_a}\,\lambda \right ) \right ) ^{n}}{a}}{d{\it \_a}}-y+\int \!{\frac {b \left ( \arctan \left ( x\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}x \right ) \lambda \right ) \right ) ^{-n}}{d{\it \_b}}+{\it \_F1} \left ( -y+\int \!{\frac {b \left ( \arctan \left ( x\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}x,-\int ^{y}\! \left ( \arctan \left ( \RootOf \left ( {\it \_b}-\int ^{{\it \_Z}}\!{\frac {b \left ( \arctan \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}}{a}}{d{\it \_b}}-y+\int \!{\frac {b \left ( \arctan \left ( x\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}x \right ) \lambda \right ) \right ) ^{-n}{d{\it \_b}}+\int \!{\frac { \left ( \arctan \left ( \beta \,z \right ) \right ) ^{-k}b}{c}}\,{\rm d}z \right ) \]
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