Added April 13, 2019.
Problem Chapter 5.7.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = w + c_1 \arctan ^k(\lambda x) + c_2 \arctan ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == w[x,y]+ c1*ArcTan[lambda*x]^k+c2*ArcTan[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {x}{a}} \left (\int _1^x\frac {e^{-\frac {K[1]}{a}} \left (\text {c1} \tan ^{-1}(\lambda K[1])^k+\text {c2} \tan ^{-1}\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )^n\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = w(x,y)+c1*arctan(lambda*x)^k+c2*arctan(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\mathit {c1} \arctan \left (\textit {\_a} \lambda \right )^{k}+\mathit {c2} \arctan \left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \beta }{a}\right )^{n}\right ) {\mathrm e}^{-\frac {\textit {\_a}}{a}}}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {x}{a}}\]
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Added April 13, 2019.
Problem Chapter 5.7.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c w + \arctan ^k(\lambda x) \arctan ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+ ArcTan[lambda*x]^k*ArcTan[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} \tan ^{-1}(\lambda K[1])^k \tan ^{-1}\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )^n}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+ arctan(lambda*x)^k*arctan(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\arctan \left (\textit {\_a} \lambda \right )^{k} \arctan \left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \beta }{a}\right )^{n} {\mathrm e}^{-\frac {\textit {\_a} c}{a}}}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]
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Added April 13, 2019.
Problem Chapter 5.7.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = \left ( c_1 \arctan (\lambda _1 x) + c_2 \arctan (\lambda _2 y)\right ) w+ s_1 \arctan ^n(\beta _1 x)+ s_2 \arctan ^k(\beta _2 y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == ( c1*ArcTan[lambda1*x] + c2*ArcTan[lambda2*y])*w[x,y]+ s1*ArcTan[beta1*x]^n+ s2*ArcTan[beta2*y]^k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \left (\text {lambda1}^2 x^2+1\right )^{-\frac {\text {c1}}{2 a \text {lambda1}}} \exp \left (\frac {\text {c2} \left (2 \text {lambda2} y \tan ^{-1}(\text {lambda2} y)-\log \left (a^2 \left (\text {lambda2}^2 y^2+1\right )\right )\right )}{2 b \text {lambda2}}+\frac {\text {c1} x \tan ^{-1}(\text {lambda1} x)}{a}\right ) \left (\int _1^x\frac {\exp \left (-\frac {b \text {c1} \tan ^{-1}(\text {lambda1} K[1]) K[1]+\text {c2} \tan ^{-1}\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right ) (-b x+a y+b K[1])}{a b}\right ) \left (\text {s2} \tan ^{-1}\left (\text {beta2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )^k+\text {s1} \tan ^{-1}(\text {beta1} K[1])^n\right ) \left (\text {lambda1}^2 K[1]^2+1\right )^{\frac {\text {c1}}{2 a \text {lambda1}}} \left (\left (\text {lambda2}^2 y^2+1\right ) a^2+2 b \text {lambda2}^2 y (K[1]-x) a+b^2 \text {lambda2}^2 (x-K[1])^2\right )^{\frac {\text {c2}}{2 b \text {lambda2}}}}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = ( c1*arctan(lambda1*x) + c2*arctan(lambda2*y))*w(x,y)+ s1*arctan(beta1*x)^n+ s2*arctan(beta2*y)^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\mathit {s1} \arctan \left (\textit {\_a} \beta 1 \right )^{n}+\mathit {s2} \arctan \left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \beta 2}{a}\right )^{k}\right ) \left (\frac {a^{2}+\left (a y -\left (-\textit {\_a} +x \right ) b \right )^{2} \lambda 2^{2}}{a^{2}}\right )^{\frac {\mathit {c2}}{2 b \lambda 2}} \left (\textit {\_a}^{2} \lambda 1^{2}+1\right )^{\frac {\mathit {c1}}{2 a \lambda 1}} {\mathrm e}^{\frac {-\textit {\_a} b \mathit {c1} \arctan \left (\textit {\_a} \lambda 1 \right )-\left (a y +\left (\textit {\_a} -x \right ) b \right ) \mathit {c2} \arctan \left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \lambda 2}{a}\right )}{a b}}}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) \left (\lambda 1^{2} x^{2}+1\right )^{-\frac {\mathit {c1}}{2 a \lambda 1}} \left (\lambda 2^{2} y^{2}+1\right )^{-\frac {\mathit {c2}}{2 b \lambda 2}} {\mathrm e}^{\frac {a \mathit {c2} y \arctan \left (\lambda 2 y \right )+b \mathit {c1} x \arctan \left (\lambda 1 x \right )}{a b}}\]
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Added April 13, 2019.
Problem Chapter 5.7.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arctan ^m(\mu x) w_y = c \arctan ^k(\nu x) w + p \arctan ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*ArcTan[mu*x]^m*D[w[x, y], y] == c*ArcTan[nu*x]^k*w[x,y]+p*ArcTan[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \tan ^{-1}(\nu K[2])^k}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \tan ^{-1}(\nu K[2])^k}{a}dK[2]\right ) p \tan ^{-1}\left (\beta \left (y-\int _1^x\frac {b \tan ^{-1}(\mu K[1])^m}{a}dK[1]+\int _1^{K[3]}\frac {b \tan ^{-1}(\mu K[1])^m}{a}dK[1]\right )\right ){}^n}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \tan ^{-1}(\mu K[1])^m}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*arctan(mu*x)^m*diff(w(x,y),y) = c*arctan(nu*x)^k*w(x,y)+p*arctan(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {p \arctan \left (\frac {\left (b \left (\int \arctan \left (\textit {\_f} \mu \right )^{m}d \textit {\_f} \right )+\left (y -\left (\int \frac {b \arctan \left (\mu x \right )^{m}}{a}d x \right )\right ) a \right ) \beta }{a}\right )^{n} {\mathrm e}^{-\frac {c \left (\int \arctan \left (\textit {\_f} \nu \right )^{k}d \textit {\_f} \right )}{a}}}{a}d \textit {\_f} +\textit {\_F1} \left (y -\left (\int \frac {b \arctan \left (\mu x \right )^{m}}{a}d x \right )\right )\right ) {\mathrm e}^{\int \frac {c \arctan \left (\nu x \right )^{k}}{a}d x}\]
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Added April 13, 2019.
Problem Chapter 5.7.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arctan ^m(\mu x) w_y = c \arctan ^k(\nu y) w + p \arctan ^n(\beta x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*ArcTan[mu*x]^m*D[w[x, y], y] == c*ArcTan[nu*y]^k*w[x,y]+p*ArcTan[beta*x]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \tan ^{-1}\left (\nu \left (y-\int _1^x\frac {b \tan ^{-1}(\mu K[1])^m}{a}dK[1]+\int _1^{K[2]}\frac {b \tan ^{-1}(\mu K[1])^m}{a}dK[1]\right )\right ){}^k}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \tan ^{-1}\left (\nu \left (y-\int _1^x\frac {b \tan ^{-1}(\mu K[1])^m}{a}dK[1]+\int _1^{K[2]}\frac {b \tan ^{-1}(\mu K[1])^m}{a}dK[1]\right )\right ){}^k}{a}dK[2]\right ) p \tan ^{-1}(\beta K[3])^n}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \tan ^{-1}(\mu K[1])^m}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*arctan(mu*x)^m*diff(w(x,y),y) = c*arctan(nu*y)^k*w(x,y)+p*arctan(beta*x)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {p \arctan \left (\textit {\_f} \beta \right )^{n} {\mathrm e}^{-\frac {c \left (\int \arctan \left (\frac {\left (b \left (\int \arctan \left (\textit {\_f} \mu \right )^{m}d \textit {\_f} \right )+\left (y -\left (\int \frac {b \arctan \left (\mu x \right )^{m}}{a}d x \right )\right ) a \right ) \nu }{a}\right )^{k}d \textit {\_f} \right )}{a}}}{a}d \textit {\_f} +\textit {\_F1} \left (y -\left (\int \frac {b \arctan \left (\mu x \right )^{m}}{a}d x \right )\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {c \arctan \left (\left (y +\int \frac {b \arctan \left (\textit {\_b} \mu \right )^{m}}{a}d \textit {\_b} -\left (\int \frac {b \arctan \left (\mu x \right )^{m}}{a}d x \right )\right ) \nu \right )^{k}}{a}d \textit {\_b}}\]
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