Added March 9, 2019.
Problem Chapter 4.7.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = \left ( c \arccot (\frac {x}{\lambda } + k \arccot (\frac {y}{\beta } ) \right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == (c*ArcCot[x/lambda] + k*ArcCot[y/beta])*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \left (\lambda ^2+x^2\right )^{\frac {c \lambda }{2 a}} c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {k \left (a \beta \log \left (a^2 \left (\beta ^2+y^2\right )\right )+2 \tan ^{-1}\left (\frac {y}{\beta }\right ) (b x-a y)+2 b x \cot ^{-1}\left (\frac {y}{\beta }\right )\right )+2 b c x \cot ^{-1}\left (\frac {x}{\lambda }\right )}{2 a b}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = (c*arccot(x/lambda)+k*arccot(y/beta))*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {\beta ^{2}+y^{2}}{\beta ^{2}}\right )^{\frac {\beta k}{2 b}} \left (\frac {x^{2}}{\lambda ^{2}}+1\right )^{\frac {c \lambda }{2 a}} \textit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\frac {-2 a k y \arctan \left (\frac {y}{\beta }\right )-2 \left (c \arctan \left (\frac {x}{\lambda }\right )-\frac {\pi \left (c +k \right )}{2}\right ) b x}{2 a b}}\]
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Added March 9, 2019.
Problem Chapter 4.7.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c \arccot (\lambda x+\beta y) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*ArcCot[lambda*x + beta*y]*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {c \left (a \log \left (a^2 \left (\beta ^2 y^2+2 \beta \lambda x y+\lambda ^2 x^2+1\right )\right )+2 \beta (b x-a y) \tan ^{-1}(\beta y+\lambda x)+2 x (a \lambda +b \beta ) \cot ^{-1}(\beta y+\lambda x)\right )}{2 a (a \lambda +b \beta )}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*arccot(lambda*x+beta*y)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\beta ^{2} y^{2}+2 \beta \lambda x y +\lambda ^{2} x^{2}+1\right )^{\frac {c}{2 a \lambda +2 b \beta }} \textit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\frac {\left (-2 \left (\beta y +\lambda x \right ) a \arctan \left (\beta y +\lambda x \right )+\pi \left (a \lambda +b \beta \right ) x \right ) c}{2 \left (a \lambda +b \beta \right ) a}}\]
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Added March 9, 2019.
Problem Chapter 4.7.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = a x \arccot (\lambda x+\beta y) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == a*x*ArcCot[lambda*x + beta*y]*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {1}{4} \left (2 x^2 \cot ^{-1}(\beta y+\lambda x)+\frac {i (i a \beta y+a-i b \beta x)^2 \log (-a (\beta y+\lambda x-i))+i (b \beta x-a (\beta y+i))^2 \log (a (\beta y+\lambda x+i))+2 a x (a \lambda +b \beta )}{(a \lambda +b \beta )^2}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = a*x*arccot(lambda*x+beta*y)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\beta ^{2} y^{2}+2 \beta \lambda x y +\lambda ^{2} x^{2}+1\right )^{-\frac {\left (a y -b x \right ) a \beta }{2 \left (a \lambda +b \beta \right )^{2}}} \textit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\frac {2 \pi a b \beta \lambda \,x^{2}+\pi \,b^{2} \beta ^{2} x^{2}+\left (\pi \,\lambda ^{2} x^{2}+2 \beta y +2 \lambda x \right ) a^{2}-2 \left (2 \left (\beta y +\lambda x \right ) b \beta x +\left (-\beta ^{2} y^{2}+\lambda ^{2} x^{2}+1\right ) a \right ) a \arctan \left (\beta y +\lambda x \right )}{4 \left (a \lambda +b \beta \right )^{2}}}\]
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Added March 9, 2019.
Problem Chapter 4.7.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arccot ^n(\lambda x)w_y = \left ( c \arccot ^m(\mu x) + s \arccot ^k(\beta y) \right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*ArcCot[lambda*x]^n*D[w[x, y], y] == (c*ArcCot[mu*x]^m + s*ArcCot[beta*y]^k)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\int _1^x\frac {b \cot ^{-1}(\lambda K[1])^n}{a}dK[1]\right ) \exp \left (\int _1^x\frac {s \cot ^{-1}\left (\beta \left (y-\int _1^x\frac {b \cot ^{-1}(\lambda K[1])^n}{a}dK[1]+\int _1^{K[2]}\frac {b \cot ^{-1}(\lambda K[1])^n}{a}dK[1]\right )\right ){}^k+c \cot ^{-1}(\mu K[2])^m}{a}dK[2]\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*arccot(lambda*x)^n*diff(w(x,y),y) =(c*arccot(mu*x)^m+s*arccot(beta*y)^k)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (y -\left (\int \frac {b \left (-\arctan \left (\lambda x \right )+\frac {\pi }{2}\right )^{n}}{a}d x \right )\right ) {\mathrm e}^{\int _{}^{x}\frac {c \left (-\arctan \left (\textit {\_b} \mu \right )+\frac {\pi }{2}\right )^{m}+s \left (-\arctan \left (\left (y +\int \frac {b \left (-\arctan \left (\textit {\_b} \lambda \right )+\frac {\pi }{2}\right )^{n}}{a}d \textit {\_b} -\left (\int \frac {b \left (-\arctan \left (\lambda x \right )+\frac {\pi }{2}\right )^{n}}{a}d x \right )\right ) \beta \right )+\frac {\pi }{2}\right )^{k}}{a}d \textit {\_b}}\]
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Added March 9, 2019.
Problem Chapter 4.7.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arccot ^n(\lambda y)w_y = \left ( c \arccot ^m(\mu x) + s \arccot ^k(\beta y) \right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*ArcCot[lambda*y]^n*D[w[x, y], y] == (c*ArcCot[mu*x]^m + s*ArcCot[beta*y]^k)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\int _1^y\cot ^{-1}(\lambda K[1])^{-n}dK[1]-\frac {b x}{a}\right ) \exp \left (\int _1^y\frac {\cot ^{-1}(\lambda K[2])^{-n} \left (s \cot ^{-1}(\beta K[2])^k+c \cot ^{-1}\left (\frac {\mu \left (b x-a \int _1^y\cot ^{-1}(\lambda K[1])^{-n}dK[1]+a \int _1^{K[2]}\cot ^{-1}(\lambda K[1])^{-n}dK[1]\right )}{b}\right ){}^m\right )}{b}dK[2]\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*arccot(lambda*y)^n*diff(w(x,y),y) =(c*arccot(mu*x)^m+s*arccot(beta*y)^k)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (-\frac {a \left (\int \left (-\arctan \left (\lambda y \right )+\frac {\pi }{2}\right )^{-n}d y \right )}{b}+x \right ) {\mathrm e}^{\int _{}^{y}\frac {\left (c \left (-\arctan \left (\left (-\frac {a \left (\int \left (-\arctan \left (\lambda y \right )+\frac {\pi }{2}\right )^{-n}d y \right )}{b}+x +\int \frac {a \left (-\arctan \left (\textit {\_b} \lambda \right )+\frac {\pi }{2}\right )^{-n}}{b}d \textit {\_b} \right ) \mu \right )+\frac {\pi }{2}\right )^{m}+s \left (-\arctan \left (\textit {\_b} \beta \right )+\frac {\pi }{2}\right )^{k}\right ) \mathrm {arccot}\left (\textit {\_b} \lambda \right )^{-n}}{b}d \textit {\_b}}\]
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