Added Feb. 11, 2019.
Problem Chapter 3.7.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c \arccot \frac {x}{\lambda }+ k \arccot \frac {y}{\beta } \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*ArcCot[x/lambda] + k*ArcCot[y/beta]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \frac {2 a b c_1\left (y-\frac {b x}{a}\right )+a \beta k \log \left (a^2 \left (\beta ^2+y^2\right )\right )-2 a k y \tan ^{-1}\left (\frac {y}{\beta }\right )+2 b k x \tan ^{-1}\left (\frac {y}{\beta }\right )+2 b k x \cot ^{-1}\left (\frac {y}{\beta }\right )+b c \lambda \log \left (\lambda ^2+x^2\right )+2 b c x \cot ^{-1}\left (\frac {x}{\lambda }\right )}{2 a b}\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x) + b*diff(w(x,y),y) = c*arccot(x/lambda)+k*arccot(y/beta); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \frac {a \beta k \ln \left (\frac {\beta ^{2}+y^{2}}{\beta ^{2}}\right )-2 a k y \arctan \left (\frac {y}{\beta }\right )+b c \lambda \ln \left (\frac {x^{2}}{\lambda ^{2}}+1\right )-2 b c x \arctan \left (\frac {x}{\lambda }\right )+2 a b \mathit {\_F1} \left (\frac {a y -b x}{a}\right )+\pi \left (c +k \right ) b x}{2 a b}\]
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Added Feb. 11, 2019.
Problem Chapter 3.7.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c \arccot (\lambda x+\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*ArcCot[lambda*x + beta*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \frac {c \left (a \log \left (a^2 \left (\beta ^2 y^2+2 \beta \lambda x y+\lambda ^2 x^2+1\right )\right )+2 \beta (b x-a y) \tan ^{-1}(\beta y+\lambda x)+2 x (a \lambda +b \beta ) \cot ^{-1}(\beta y+\lambda x)\right )}{2 a (a \lambda +b \beta )}+c_1\left (y-\frac {b x}{a}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x) + b*diff(w(x,y),y) = c *arccot(lambda*x+beta*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \frac {a c \ln \left (\beta ^{2} y^{2}+2 \beta \lambda x y +\lambda ^{2} x^{2}+1\right )-2 \left (\left (\beta y +\lambda x \right ) a \arctan \left (\beta y +\lambda x \right )-\frac {\pi \left (a \lambda +b \beta \right ) x}{2}\right ) c +\left (2 a^{2} \lambda +2 a b \beta \right ) \mathit {\_F1} \left (\frac {a y -b x}{a}\right )}{2 \left (a \lambda +b \beta \right ) a}\]
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Added Feb. 11, 2019.
Problem Chapter 3.7.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x w_x + y w_y = a x \arccot (\lambda x+\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + y*D[w[x, y], y] == a*x*ArcCot[lambda*x + beta*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to a x \left (\frac {\log \left (\beta ^2 y^2+2 \beta \lambda x y+\lambda ^2 x^2+1\right )}{2 \beta y+2 \lambda x}+\cot ^{-1}(\beta y+\lambda x)\right )+c_1\left (\frac {y}{x}\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x) + y*diff(w(x,y),y) = a*x *arccot(lambda*x+beta*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \frac {a x \ln \left (\left (\frac {\beta y}{x}+\lambda \right )^{2} x^{2}+1\right )+\left (\beta y +\lambda x \right ) \left (-2 a x \arctan \left (\beta y +\lambda x \right )+\pi a x +2 \mathit {\_F1} \left (\frac {y}{x}\right )\right )}{2 \beta y +2 \lambda x}\]
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Added Feb. 11, 2019.
Problem Chapter 3.7.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b \arccot ^n(\lambda x) w_y = c \arccot ^m(\mu x)+s \arccot ^k(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*ArcCot[lambda*x]^n*D[w[x, y], y] == a*ArcCot[mu*x]^m + ArcCot[beta*y]^k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^x\left (\frac {\cot ^{-1}\left (\beta \left (y-\int _1^x\frac {b \cot ^{-1}(\lambda K[1])^n}{a}dK[1]+\int _1^{K[2]}\frac {b \cot ^{-1}(\lambda K[1])^n}{a}dK[1]\right )\right ){}^k}{a}+\cot ^{-1}(\mu K[2])^m\right )dK[2]+c_1\left (y-\int _1^x\frac {b \cot ^{-1}(\lambda K[1])^n}{a}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x) + b*arccot(lambda*x)*diff(w(x,y),y) = a*arccot(mu*x)^m+arccot(beta*y)^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \int _{}^{x}\left (\left (-\arctan \left (\mathit {\_a} \mu \right )+\frac {\pi }{2}\right )^{m}+\frac {\left (-\arctan \left (\frac {\left (\frac {b \ln \left (\mathit {\_a}^{2} \lambda ^{2}+1\right )}{2}-\frac {b \ln \left (\lambda ^{2} x^{2}+1\right )}{2}+\left (-\mathit {\_a} b \arctan \left (\mathit {\_a} \lambda \right )+b x \arctan \left (\lambda x \right )+a y -\frac {\pi \left (-\mathit {\_a} +x \right ) b}{2}\right ) \lambda \right ) \beta }{a \lambda }\right )+\frac {\pi }{2}\right )^{k}}{a}\right )d\mathit {\_a} +\mathit {\_F1} \left (\frac {2 b \lambda x \arctan \left (\lambda x \right )+2 a \lambda y -\pi b \lambda x -b \ln \left (\lambda ^{2} x^{2}+1\right )}{2 a \lambda }\right )\]
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Added Feb. 11, 2019.
Problem Chapter 3.7.4.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b \arccot ^n(\lambda y) w_y = c \arccot ^m(\mu x)+s \arccot ^k(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*ArcCot[lambda*y]^n*D[w[x, y], y] == a*ArcCot[mu*x]^m + ArcCot[beta*y]^k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^y\frac {\cot ^{-1}(\lambda K[2])^{-n} \left (\cot ^{-1}(\beta K[2])^k+a \cot ^{-1}\left (\frac {\mu \left (b x-a \int _1^y\cot ^{-1}(\lambda K[1])^{-n}dK[1]+a \int _1^{K[2]}\cot ^{-1}(\lambda K[1])^{-n}dK[1]\right )}{b}\right ){}^m\right )}{b}dK[2]+c_1\left (\int _1^y\cot ^{-1}(\lambda K[1])^{-n}dK[1]-\frac {b x}{a}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x) + b*arccot(lambda*y)*diff(w(x,y),y) = a*arccot(mu*x)^m+arccot(beta*y)^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \int _{}^{y}\frac {a \left (\arctan \left (\frac {2 a \mu \left (\int \frac {1}{2 \arctan \left (\mathit {\_b} \lambda \right )-\pi }d \mathit {\_b} \right )}{b}-\left (x -\left (\int \frac {2 a}{\left (-2 \arctan \left (\lambda y \right )+\pi \right ) b}d y \right )\right ) \mu \right )+\frac {\pi }{2}\right )^{m}+\left (-\arctan \left (\mathit {\_b} \beta \right )+\frac {\pi }{2}\right )^{k}}{b \,\mathrm {arccot}\left (\mathit {\_b} \lambda \right )}d\mathit {\_b} +\mathit {\_F1} \left (x -\left (\int \frac {2 a}{\left (-2 \arctan \left (\lambda y \right )+\pi \right ) b}d y \right )\right )\]
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