Added March 9, 2019.
Problem Chapter 4.7.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = \left ( c \arctan (\frac {x}{\lambda } + k \arctan (\frac {y}{\beta } ) \right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == (c*ArcTan[x/lambda] + k*ArcTan[y/beta])*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \left (\lambda ^2+x^2\right )^{-\frac {c \lambda }{2 a}} c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {k \left (2 y \tan ^{-1}\left (\frac {y}{\beta }\right )-\beta \log \left (a^2 \left (\beta ^2+y^2\right )\right )\right )}{2 b}+\frac {c x \tan ^{-1}\left (\frac {x}{\lambda }\right )}{a}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = (c*arctan(x/lambda)+k*arctan(y/beta))*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {\beta ^{2}+y^{2}}{\beta ^{2}}\right )^{-\frac {\beta k}{2 b}} \left (\frac {x^{2}}{\lambda ^{2}}+1\right )^{-\frac {c \lambda }{2 a}} \mathit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\frac {a k y \arctan \left (\frac {y}{\beta }\right )+b c x \arctan \left (\frac {x}{\lambda }\right )}{a b}}\]
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Added March 9, 2019.
Problem Chapter 4.7.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c \arctan (\lambda x+\beta y) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*ArcTan[lambda*x + beta*y]*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {c \left (2 (\beta y+\lambda x) \tan ^{-1}(\beta y+\lambda x)-\log \left (a^2 \left (\beta ^2 y^2+2 \beta \lambda x y+\lambda ^2 x^2+1\right )\right )\right )}{2 (a \lambda +b \beta )}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*arctan(lambda*x+beta*y)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\beta ^{2} y^{2}+2 \beta \lambda x y +\lambda ^{2} x^{2}+1\right )^{-\frac {c}{2 a \lambda +2 b \beta }} \mathit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\frac {\left (\beta y +\lambda x \right ) c \arctan \left (\beta y +\lambda x \right )}{a \lambda +b \beta }}\]
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Added March 9, 2019.
Problem Chapter 4.7.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = a x \arctan (\lambda x+\beta y) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == a*x*ArcTan[lambda*x + beta*y]*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {1}{4} \left (2 x^2 \tan ^{-1}(\beta y+\lambda x)+\frac {i (a+i \beta (b x-a y))^2 \log (a (\beta y+\lambda x+i))+i (b \beta x+a (-\beta y+i))^2 \log (-a (\beta y+\lambda x-i))-2 a x (a \lambda +b \beta )}{(a \lambda +b \beta )^2}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = a*x*arctan(lambda*x+beta*y)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\beta ^{2} y^{2}+2 \beta \lambda x y +\lambda ^{2} x^{2}+1\right )^{\frac {\left (a y -b x \right ) a \beta }{2 \left (a \lambda +b \beta \right )^{2}}} \mathit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\frac {\left (-\left (\beta y +\lambda x \right ) a +\left (2 \left (\beta y +\lambda x \right ) b \beta x +\left (-\beta ^{2} y^{2}+\lambda ^{2} x^{2}+1\right ) a \right ) \arctan \left (\beta y +\lambda x \right )\right ) a}{2 \left (a \lambda +b \beta \right )^{2}}}\]
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Added March 9, 2019.
Problem Chapter 4.7.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arctan ^n(\lambda x)w_y = \left ( c \arctan ^m(\mu x) + s \arctan ^k(\beta y) \right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*ArcTan[lambda*x]^n*D[w[x, y], y] == (c*ArcTan[mu*x]^m + s*ArcTan[beta*y]^k)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\int _1^x\frac {b \tan ^{-1}(\lambda K[1])^n}{a}dK[1]\right ) \exp \left (\int _1^x\frac {s \tan ^{-1}\left (\beta \left (y-\int _1^x\frac {b \tan ^{-1}(\lambda K[1])^n}{a}dK[1]+\int _1^{K[2]}\frac {b \tan ^{-1}(\lambda K[1])^n}{a}dK[1]\right )\right ){}^k+c \tan ^{-1}(\mu K[2])^m}{a}dK[2]\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*arctan(lambda*x)^n*diff(w(x,y),y) =(c*arctan(mu*x)^m+s*arctan(beta*y)^k)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (y -\left (\int \frac {b \arctan \left (\lambda x \right )^{n}}{a}d x \right )\right ) {\mathrm e}^{\int _{}^{x}\frac {c \arctan \left (\mathit {\_b} \mu \right )^{m}+s \arctan \left (\left (y +\int \frac {b \arctan \left (\mathit {\_b} \lambda \right )^{n}}{a}d \mathit {\_b} -\left (\int \frac {b \arctan \left (\lambda x \right )^{n}}{a}d x \right )\right ) \beta \right )^{k}}{a}d\mathit {\_b}}\]
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Added March 9, 2019.
Problem Chapter 4.7.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arctan ^n(\lambda y)w_y = \left ( c \arctan ^m(\mu x) + s \arctan ^k(\beta y) \right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*ArcTan[lambda*y]^n*D[w[x, y], y] == (c*ArcTan[mu*x]^m + s*ArcTan[beta*y]^k)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\int _1^y\tan ^{-1}(\lambda K[1])^{-n}dK[1]-\frac {b x}{a}\right ) \exp \left (\int _1^y\frac {\tan ^{-1}(\lambda K[2])^{-n} \left (s \tan ^{-1}(\beta K[2])^k+c \tan ^{-1}\left (\frac {\mu \left (b x-a \int _1^y\tan ^{-1}(\lambda K[1])^{-n}dK[1]+a \int _1^{K[2]}\tan ^{-1}(\lambda K[1])^{-n}dK[1]\right )}{b}\right ){}^m\right )}{b}dK[2]\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*arctan(lambda*y)^n*diff(w(x,y),y) =(c*arctan(mu*x)^m+s*arctan(beta*y)^k)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (-\frac {a \left (\int \arctan \left (\lambda y \right )^{-n}d y \right )}{b}+x \right ) {\mathrm e}^{\int _{}^{y}\frac {\left (c \arctan \left (\left (-\frac {a \left (\int \arctan \left (\lambda y \right )^{-n}d y \right )}{b}+x +\int \frac {a \arctan \left (\mathit {\_b} \lambda \right )^{-n}}{b}d \mathit {\_b} \right ) \mu \right )^{m}+s \arctan \left (\mathit {\_b} \beta \right )^{k}\right ) \arctan \left (\mathit {\_b} \lambda \right )^{-n}}{b}d\mathit {\_b}}\]
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