Added April 13, 2019.
Problem Chapter 5.7.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = w + c_1 \arccot ^k(\lambda x) + c_2 \arccot ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == w[x,y]+ c1*ArcCot[lambda*x]^k+c2*ArcCot[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {x}{a}} \left (\int _1^x\frac {e^{-\frac {K[1]}{a}} \left (\text {c1} \cot ^{-1}(\lambda K[1])^k+\text {c2} \cot ^{-1}\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )^n\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = w(x,y)+c1*arccot(lambda*x)^k+c2*arccot(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\mathit {c1} \left (-\arctan \left (\textit {\_a} \lambda \right )+\frac {\pi }{2}\right )^{k}+\mathit {c2} \left (-\arctan \left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \beta }{a}\right )+\frac {\pi }{2}\right )^{n}\right ) {\mathrm e}^{-\frac {\textit {\_a}}{a}}}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {x}{a}}\]
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Added April 13, 2019.
Problem Chapter 5.7.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c w + \arccot ^k(\lambda x) \arccot ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+ ArcCot[lambda*x]^k*ArcCot[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} \cot ^{-1}(\lambda K[1])^k \cot ^{-1}\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )^n}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+ arccot(lambda*x)^k*arccot(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (-\arctan \left (\textit {\_a} \lambda \right )+\frac {\pi }{2}\right )^{k} \left (-\arctan \left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \beta }{a}\right )+\frac {\pi }{2}\right )^{n} {\mathrm e}^{-\frac {\textit {\_a} c}{a}}}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]
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Added April 13, 2019.
Problem Chapter 5.7.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = \left ( c_1 \arccot (\lambda _1 x) + c_2 \arccot (\lambda _2 y)\right ) w+ s_1 \arccot ^n(\beta _1 x)+ s_2 \arccot ^k(\beta _2 y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == ( c1*ArcCot[lambda1*x] + c2*ArcCot[lambda2*y])*w[x,y]+ s1*ArcCot[beta1*x]^n+ s2*ArcCot[beta2*y]^k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \left (\text {lambda1}^2 x^2+1\right )^{\frac {\text {c1}}{2 a \text {lambda1}}} \exp \left (\frac {\frac {\text {c2} \left (a \log \left (a^2 \left (\text {lambda2}^2 y^2+1\right )\right )+2 \text {lambda2} \tan ^{-1}(\text {lambda2} y) (b x-a y)+2 b \text {lambda2} x \cot ^{-1}(\text {lambda2} y)\right )}{b \text {lambda2}}+2 \text {c1} x \cot ^{-1}(\text {lambda1} x)}{2 a}\right ) \left (\int _1^x\frac {\exp \left (-\frac {\text {c2} (b x-a y) \tan ^{-1}\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )+b \left (\text {c1} \cot ^{-1}(\text {lambda1} K[1])+\text {c2} \cot ^{-1}\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )\right ) K[1]}{a b}\right ) \left (\text {s2} \cot ^{-1}\left (\text {beta2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )^k+\text {s1} \cot ^{-1}(\text {beta1} K[1])^n\right ) \left (\text {lambda1}^2 K[1]^2+1\right )^{-\frac {\text {c1}}{2 a \text {lambda1}}} \left (\left (\text {lambda2}^2 y^2+1\right ) a^2+2 b \text {lambda2}^2 y (K[1]-x) a+b^2 \text {lambda2}^2 (x-K[1])^2\right )^{-\frac {\text {c2}}{2 b \text {lambda2}}}}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = ( c1*arccot(lambda1*x) + c2*arccot(lambda2*y))*w(x,y)+ s1*arccot(beta1*x)^n+ s2*arccot(beta2*y)^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\mathit {s1} \left (-\arctan \left (\textit {\_a} \beta 1 \right )+\frac {\pi }{2}\right )^{n}+\mathit {s2} \left (-\arctan \left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \beta 2}{a}\right )+\frac {\pi }{2}\right )^{k}\right ) \left (\frac {a^{2}+\left (a y -\left (-\textit {\_a} +x \right ) b \right )^{2} \lambda 2^{2}}{a^{2}}\right )^{-\frac {\mathit {c2}}{2 b \lambda 2}} \left (\textit {\_a}^{2} \lambda 1^{2}+1\right )^{-\frac {\mathit {c1}}{2 a \lambda 1}} {\mathrm e}^{\frac {2 \left (\mathit {c1} \arctan \left (\textit {\_a} \lambda 1 \right )-\frac {\pi \left (\mathit {c1} +\mathit {c2} \right )}{2}\right ) \textit {\_a} b +2 \left (a y +\left (\textit {\_a} -x \right ) b \right ) \mathit {c2} \arctan \left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \lambda 2}{a}\right )}{2 a b}}}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) \left (\lambda 1^{2} x^{2}+1\right )^{\frac {\mathit {c1}}{2 a \lambda 1}} \left (\lambda 2^{2} y^{2}+1\right )^{\frac {\mathit {c2}}{2 b \lambda 2}} {\mathrm e}^{\frac {-2 a \mathit {c2} y \arctan \left (\lambda 2 y \right )-2 \left (\mathit {c1} \arctan \left (\lambda 1 x \right )-\frac {\pi \left (\mathit {c1} +\mathit {c2} \right )}{2}\right ) b x}{2 a b}}\]
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Added April 13, 2019.
Problem Chapter 5.7.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arccot ^m(\mu x) w_y = c \arccot ^k(\nu x) w + p \arccot ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*ArcCot[mu*x]^m*D[w[x, y], y] == c*ArcCot[nu*x]^k*w[x,y]+p*ArcCot[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \cot ^{-1}(\nu K[2])^k}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \cot ^{-1}(\nu K[2])^k}{a}dK[2]\right ) p \cot ^{-1}\left (\beta \left (y-\int _1^x\frac {b \cot ^{-1}(\mu K[1])^m}{a}dK[1]+\int _1^{K[3]}\frac {b \cot ^{-1}(\mu K[1])^m}{a}dK[1]\right )\right ){}^n}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \cot ^{-1}(\mu K[1])^m}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*arccot(mu*x)^m*diff(w(x,y),y) = c*arccot(nu*x)^k*w(x,y)+p*arccot(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {p \left (-\arctan \left (\frac {\left (b \left (\int \left (-\arctan \left (\textit {\_f} \mu \right )+\frac {\pi }{2}\right )^{m}d \textit {\_f} \right )+\left (y -\left (\int \frac {b \left (-\arctan \left (\mu x \right )+\frac {\pi }{2}\right )^{m}}{a}d x \right )\right ) a \right ) \beta }{a}\right )+\frac {\pi }{2}\right )^{n} {\mathrm e}^{-\frac {c \left (\int \left (-\arctan \left (\textit {\_f} \nu \right )+\frac {\pi }{2}\right )^{k}d \textit {\_f} \right )}{a}}}{a}d \textit {\_f} +\textit {\_F1} \left (y -\left (\int \frac {b \left (-\arctan \left (\mu x \right )+\frac {\pi }{2}\right )^{m}}{a}d x \right )\right )\right ) {\mathrm e}^{\int \frac {c \left (-\arctan \left (\nu x \right )+\frac {\pi }{2}\right )^{k}}{a}d x}\]
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Added April 13, 2019.
Problem Chapter 5.7.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arccot ^m(\mu x) w_y = c \arccot ^k(\nu y) w + p \arccot ^n(\beta x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*ArcCot[mu*x]^m*D[w[x, y], y] == c*ArcCot[nu*y]^k*w[x,y]+p*ArcCot[beta*x]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \cot ^{-1}\left (\nu \left (y-\int _1^x\frac {b \cot ^{-1}(\mu K[1])^m}{a}dK[1]+\int _1^{K[2]}\frac {b \cot ^{-1}(\mu K[1])^m}{a}dK[1]\right )\right ){}^k}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \cot ^{-1}\left (\nu \left (y-\int _1^x\frac {b \cot ^{-1}(\mu K[1])^m}{a}dK[1]+\int _1^{K[2]}\frac {b \cot ^{-1}(\mu K[1])^m}{a}dK[1]\right )\right ){}^k}{a}dK[2]\right ) p \cot ^{-1}(\beta K[3])^n}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \cot ^{-1}(\mu K[1])^m}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*arccot(mu*x)^m*diff(w(x,y),y) = c*arccot(nu*y)^k*w(x,y)+p*arccot(beta*x)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {p \left (-\arctan \left (\textit {\_f} \beta \right )+\frac {\pi }{2}\right )^{n} {\mathrm e}^{-\frac {c \left (\int \left (-\arctan \left (\frac {\left (b \left (\int \left (-\arctan \left (\textit {\_f} \mu \right )+\frac {\pi }{2}\right )^{m}d \textit {\_f} \right )+\left (y -\left (\int \frac {b \left (-\arctan \left (\mu x \right )+\frac {\pi }{2}\right )^{m}}{a}d x \right )\right ) a \right ) \nu }{a}\right )+\frac {\pi }{2}\right )^{k}d \textit {\_f} \right )}{a}}}{a}d \textit {\_f} +\textit {\_F1} \left (y -\left (\int \frac {b \left (-\arctan \left (\mu x \right )+\frac {\pi }{2}\right )^{m}}{a}d x \right )\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {c \left (-\arctan \left (\left (y +\int \frac {b \left (-\arctan \left (\textit {\_b} \mu \right )+\frac {\pi }{2}\right )^{m}}{a}d \textit {\_b} -\left (\int \frac {b \left (-\arctan \left (\mu x \right )+\frac {\pi }{2}\right )^{m}}{a}d x \right )\right ) \nu \right )+\frac {\pi }{2}\right )^{k}}{a}d \textit {\_b}}\]
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