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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \left ( {\frac {{x}^{2}}{{\lambda }^{2}}}+1 \right ) ^{-{\frac {\lambda \,c}{2\,a}}} \left ( {\frac {{\beta }^{2}+{y}^{2}}{{\beta }^{2}}} \right ) ^{-{\frac {k\beta }{2\,b}}}{{\rm e}^{{\frac {1}{ba} \left ( a\arctan \left ( {\frac {y}{\beta }} \right ) ky+cx\arctan \left ( {\frac {x}{\lambda }} \right ) b \right ) }}}\]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \left ( {\beta }^{2}{y}^{2}+2\,\beta \,\lambda \,xy+{x}^{2}{\lambda }^{2}+1 \right ) ^{-{\frac {c}{2\,a\lambda +2\,\beta \,b}}}{{\rm e}^{{\frac {c\arctan \left ( \beta \,y+x\lambda \right ) \left ( \beta \,y+x\lambda \right ) }{a\lambda +\beta \,b}}}}\]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \left ( {\beta }^{2}{y}^{2}+2\,\beta \,\lambda \,xy+{x}^{2}{\lambda }^{2}+1 \right ) ^{{\frac {\beta \, \left ( ay-xb \right ) a}{2\, \left ( a\lambda +\beta \,b \right ) ^{2}}}}{{\rm e}^{{\frac { \left ( \left ( \left ( -{\beta }^{2}{y}^{2}+{x}^{2}{\lambda }^{2}+1 \right ) a+2\,b\beta \,x \left ( \beta \,y+x\lambda \right ) \right ) \arctan \left ( \beta \,y+x\lambda \right ) -a \left ( \beta \,y+x\lambda \right ) \right ) a}{2\, \left ( a\lambda +\beta \,b \right ) ^{2}}}}}\]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int \!{\frac {b \left ( \arctan \left ( x\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{a} \left ( c \left ( \arctan \left ( \mu \,{\it \_b} \right ) \right ) ^{m}+s \left ( \arctan \left ( \beta \, \left ( \int \!{\frac {b \left ( \arctan \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}{\it \_b}-\int \!{\frac {b \left ( \arctan \left ( x\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) \right ) \right ) ^{k} \right ) }{d{\it \_b}}}}\]
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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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {a\int \! \left ( \arctan \left ( \lambda \,y \right ) \right ) ^{-n}\,{\rm d}y}{b}}+x \right ) {{\rm e}^{\int ^{y}\!{\frac { \left ( \arctan \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{b} \left ( c \left ( \arctan \left ( \mu \, \left ( \int \!{\frac { \left ( \arctan \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}a}{b}}\,{\rm d}{\it \_b}-{\frac {a\int \! \left ( \arctan \left ( \lambda \,y \right ) \right ) ^{-n}\,{\rm d}y}{b}}+x \right ) \right ) \right ) ^{m}+s \left ( \arctan \left ( \beta \,{\it \_b} \right ) \right ) ^{k} \right ) }{d{\it \_b}}}}\]
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