Added Feb. 11, 2019.
Problem Chapter 3.8.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = f(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == f[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^x\frac {f(K[1])}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x) + b*diff(w(x,y),y) = f(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \int \frac {f \left (x \right )}{a}d x +\textit {\_F1} \left (\frac {a y -b x}{a}\right )\]
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Added Feb. 11, 2019.
Problem Chapter 3.8.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = y f(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + a*D[w[x, y], y] == y*f[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^xf(K[1]) (y+a (K[1]-x))dK[1]+c_1(y-a x)\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x) + a*diff(w(x,y),y) = y*f(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \int _{}^{x}-\left (\left (-\textit {\_a} +x \right ) a -y \right ) f \left (\textit {\_a} \right )d \textit {\_a} +\textit {\_F1} \left (-a x +y \right )\]
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Added Feb. 11, 2019.
Problem Chapter 3.8.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + a w_y = y^2 f(x)+ y g(x) + h(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + a*D[w[x, y], y] == y^2*f[x] + y*g[x] + h[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^x\left (f(K[1]) (y+a (K[1]-x))^2+g(K[1]) (y+a (K[1]-x))+h(K[1])\right )dK[1]+c_1(y-a x)\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x) + a*diff(w(x,y),y) = y^2*f(x)+y*g(x)+h(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \int _{}^{x}\left (\left (\left (-\textit {\_a} +x \right ) a -y \right )^{2} f \left (\textit {\_a} \right )+\left (\left (\textit {\_a} -x \right ) a +y \right ) g \left (\textit {\_a} \right )+h \left (\textit {\_a} \right )\right )d \textit {\_a} +\textit {\_F1} \left (-a x +y \right )\]
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Added Feb. 11, 2019.
Problem Chapter 3.8.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + a w_y = y^k f(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + a*D[w[x, y], y] == y^k*f[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^xf(K[1]) (y+a (K[1]-x))^kdK[1]+c_1(y-a x)\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x) + a*diff(w(x,y),y) = y^k*f(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \int _{}^{x}\left (\left (\textit {\_a} -x \right ) a +y \right )^{k} f \left (\textit {\_a} \right )d \textit {\_a} +\textit {\_F1} \left (-a x +y \right )\]
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Added Feb. 11, 2019.
Problem Chapter 3.8.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + a w_y = e^{\lambda y} f(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + a*D[w[x, y], y] == Exp[lambda*y]*f[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^xe^{\lambda (y+a (K[1]-x))} f(K[1])dK[1]+c_1(y-a x)\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x) + a*diff(w(x,y),y) = exp(lambda*y)*f(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \int _{}^{x}{\mathrm e}^{-\left (\left (-\textit {\_a} +x \right ) a -y \right ) \lambda } f \left (\textit {\_a} \right )d \textit {\_a} +\textit {\_F1} \left (-a x +y \right )\]
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Added Feb. 11, 2019.
Problem Chapter 3.8.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + (a y + f(x) ) w_y = g(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*y + f[x])*D[w[x, y], y] == g[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^xg(K[2])dK[2]+c_1\left (y e^{-a x}-\int _1^xe^{-a K[1]} f(K[1])dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x) + (a*y+f(x))*diff(w(x,y),y) = g(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \int g \left (x \right )d x +\textit {\_F1} \left (y \,{\mathrm e}^{-a x}-\left (\int {\mathrm e}^{-a x} f \left (x \right )d x \right )\right )\]
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Added Feb. 11, 2019.
Problem Chapter 3.8.1.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + (a y + f(x) ) w_y = y^k g(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*y + f[x])*D[w[x, y], y] == y^k*g[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^xg(K[2]) \left (e^{a K[2]} \left (e^{-a x} y-\int _1^xe^{-a K[1]} f(K[1])dK[1]+\int _1^{K[2]}e^{-a K[1]} f(K[1])dK[1]\right )\right ){}^kdK[2]+c_1\left (y e^{-a x}-\int _1^xe^{-a K[1]} f(K[1])dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x) + (a*y+f(x))*diff(w(x,y),y) = y^k*g(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \int _{}^{x}\left (\left (y \,{\mathrm e}^{-a x}+\int {\mathrm e}^{-\textit {\_b} a} f \left (\textit {\_b} \right )d \textit {\_b} -\left (\int {\mathrm e}^{-a x} f \left (x \right )d x \right )\right ) {\mathrm e}^{\textit {\_b} a}\right )^{k} g \left (\textit {\_b} \right )d \textit {\_b} +\textit {\_F1} \left (y \,{\mathrm e}^{-a x}-\left (\int {\mathrm e}^{-a x} f \left (x \right )d x \right )\right )\]
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Added Feb. 11, 2019.
Problem Chapter 3.8.1.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ f(x) w_x + y^k w_y = g(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = f[x]*D[w[x, y], x] + y^k*D[w[x, y], y] == g[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^x\frac {g(K[2])}{f(K[2])}dK[2]+c_1\left (-\int _1^x\frac {1}{f(K[1])}dK[1]-\frac {y^{1-k}}{k-1}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x) + y^k*diff(w(x,y),y) = g(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \int g \left (x \right )d x +\textit {\_F1} \left (\left (\left (k -1\right ) x \,y^{k}+y \right ) y^{-k}\right )\]
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Added Feb. 11, 2019.
Problem Chapter 3.8.1.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ f(x) w_x + (y+a) w_y = b y+c \]
Mathematica ✓
ClearAll["Global`*"]; pde = f[x]*D[w[x, y], x] + (y + a)*D[w[x, y], y] == b*y + c; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^x\frac {c+b \exp \left (\int _1^{K[3]}\frac {1}{f(K[1])}dK[1]\right ) \left (\exp \left (-\int _1^x\frac {1}{f(K[1])}dK[1]\right ) y-\int _1^x\frac {a \exp \left (-\int _1^{K[2]}\frac {1}{f(K[1])}dK[1]\right )}{f(K[2])}dK[2]+\int _1^{K[3]}\frac {a \exp \left (-\int _1^{K[2]}\frac {1}{f(K[1])}dK[1]\right )}{f(K[2])}dK[2]\right )}{f(K[3])}dK[3]+c_1\left (y \exp \left (-\int _1^x\frac {1}{f(K[1])}dK[1]\right )-\int _1^x\frac {a \exp \left (-\int _1^{K[2]}\frac {1}{f(K[1])}dK[1]\right )}{f(K[2])}dK[2]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x) + (y+a)*diff(w(x,y),y) = b*y+c; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = c x +\left (\left (-x +1\right ) a +y \right ) b +\textit {\_F1} \left (\left (a +y \right ) {\mathrm e}^{-x}\right )\]
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Added Feb. 11, 2019.
Problem Chapter 3.8.1.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ f(x) w_x + (y+a x) w_y = g(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = f[x]*D[w[x, y], x] + (y + a*x)*D[w[x, y], y] == g[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^x\frac {g(K[3])}{f(K[3])}dK[3]+c_1\left (y \exp \left (-\int _1^x\frac {1}{f(K[1])}dK[1]\right )-\int _1^x\frac {a \exp \left (-\int _1^{K[2]}\frac {1}{f(K[1])}dK[1]\right ) K[2]}{f(K[2])}dK[2]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x) + (y+a*x)*diff(w(x,y),y) = g(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \int g \left (x \right )d x +\textit {\_F1} \left (\left (a x +a +y \right ) {\mathrm e}^{-x}\right )\]
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Added Feb. 11, 2019.
Problem Chapter 3.8.1.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ f(x) w_x + (y g_1(x)+g_0(x)) w_y = y^2 h_2(x)+y h_1(x) + h_0(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = f[x]*D[w[x, y], x] + (y*g1[x] + g0[x])*D[w[x, y], y] == y^2*h2[x] + y*h1[x] + h0[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^x\frac {\exp \left (2 \int _1^{K[3]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {h2}(K[3]) \left (\exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) y-\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]+\int _1^{K[3]}\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right ){}^2+\exp \left (\int _1^{K[3]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {h1}(K[3]) \left (\exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) y-\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]+\int _1^{K[3]}\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right )+\text {h0}(K[3])}{f(K[3])}dK[3]+c_1\left (y \exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right )-\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x) + (y*g1(x)+g0(x))*diff(w(x,y),y) = y^2*h2(x)+y*h1(x)+h0(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \int _{}^{x}\left (y^{2} {\mathrm e}^{2 \left (\int \mathit {g1} \left (\textit {\_f} \right )d \textit {\_f} \right )} {\mathrm e}^{\int -2 \mathit {g1} \left (x \right )d x} \mathit {h2} \left (\textit {\_f} \right )+y \,{\mathrm e}^{-\left (\int \mathit {g1} \left (x \right )d x \right )} {\mathrm e}^{\int \mathit {g1} \left (\textit {\_f} \right )d \textit {\_f}} \mathit {h1} \left (\textit {\_f} \right )+\left (\int {\mathrm e}^{-\left (\int \mathit {g1} \left (\textit {\_f} \right )d \textit {\_f} \right )} \mathit {g0} \left (\textit {\_f} \right )d \textit {\_f} \right )^{2} {\mathrm e}^{2 \left (\int \mathit {g1} \left (\textit {\_f} \right )d \textit {\_f} \right )} \mathit {h2} \left (\textit {\_f} \right )+\left (\int {\mathrm e}^{-\left (\int \mathit {g1} \left (x \right )d x \right )} \mathit {g0} \left (x \right )d x \right )^{2} {\mathrm e}^{2 \left (\int \mathit {g1} \left (\textit {\_f} \right )d \textit {\_f} \right )} \mathit {h2} \left (\textit {\_f} \right )+\left (2 y \,{\mathrm e}^{2 \left (\int \mathit {g1} \left (\textit {\_f} \right )d \textit {\_f} \right )} {\mathrm e}^{-\left (\int \mathit {g1} \left (x \right )d x \right )} \mathit {h2} \left (\textit {\_f} \right )+{\mathrm e}^{\int \mathit {g1} \left (\textit {\_f} \right )d \textit {\_f}} \mathit {h1} \left (\textit {\_f} \right )\right ) \left (\int {\mathrm e}^{-\left (\int \mathit {g1} \left (\textit {\_f} \right )d \textit {\_f} \right )} \mathit {g0} \left (\textit {\_f} \right )d \textit {\_f} \right )+\left (-2 y \,{\mathrm e}^{2 \left (\int \mathit {g1} \left (\textit {\_f} \right )d \textit {\_f} \right )} {\mathrm e}^{-\left (\int \mathit {g1} \left (x \right )d x \right )} \mathit {h2} \left (\textit {\_f} \right )-2 \left (\int {\mathrm e}^{-\left (\int \mathit {g1} \left (\textit {\_f} \right )d \textit {\_f} \right )} \mathit {g0} \left (\textit {\_f} \right )d \textit {\_f} \right ) {\mathrm e}^{2 \left (\int \mathit {g1} \left (\textit {\_f} \right )d \textit {\_f} \right )} \mathit {h2} \left (\textit {\_f} \right )-{\mathrm e}^{\int \mathit {g1} \left (\textit {\_f} \right )d \textit {\_f}} \mathit {h1} \left (\textit {\_f} \right )\right ) \left (\int {\mathrm e}^{-\left (\int \mathit {g1} \left (x \right )d x \right )} \mathit {g0} \left (x \right )d x \right )+\mathit {h0} \left (\textit {\_f} \right )\right )d \textit {\_f} +\textit {\_F1} \left (y \,{\mathrm e}^{-\left (\int \mathit {g1} \left (x \right )d x \right )}-\left (\int {\mathrm e}^{-\left (\int \mathit {g1} \left (x \right )d x \right )} \mathit {g0} \left (x \right )d x \right )\right )\]
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Added Feb. 11, 2019.
Problem Chapter 3.8.1.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ f(x) w_x + (y g_1(x)+y^k g_2(x)) w_y = h(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = f[x]*D[w[x, y], x] + (y*g1[x] + y^k*g2[x])*D[w[x, y], y] == h[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^x\frac {h(K[3])}{f(K[3])}dK[3]+c_1\left ((k-1) \int _1^x\frac {\exp \left ((k-1) \int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g2}(K[2])}{f(K[2])}dK[2]+y^{1-k} \exp \left ((k-1) \int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x) + (y*g1(x)+y^k*g2(x))*diff(w(x,y),y) = h(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \int h \left (x \right )d x +\textit {\_F1} \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {g1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {g1} \left (x \right )d x \right )} \mathit {g2} \left (x \right )d x \right )\right )\]
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Added Feb. 11, 2019.
Problem Chapter 3.8.1.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ f(x) w_x + (g_1(x)+e^{\lambda y} g_2(x)) w_y = h(x) \]
Mathematica ✗
ClearAll["Global`*"]; pde = f[x]*D[w[x, y], x] + (g1[x] + Exp[lambda*y])*D[w[x, y], y] == h[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := f(x)*diff(w(x,y),x) +(g1(x)+exp(lambda*y))*diff(w(x,y),y) = h(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \int \frac {h \left (x \right )}{f \left (x \right )}d x +\textit {\_F1} \left (\frac {-\lambda \left (\int \frac {{\mathrm e}^{\lambda \left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )}}{f \left (x \right )}d x \right )-{\mathrm e}^{-\left (y -\left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )\right ) \lambda }}{\lambda }\right )\]
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Added Feb. 11, 2019.
Problem Chapter 3.8.1.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ y^k f(x) w_x + g(x) w_y = h(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = y^k*f[x]*D[w[x, y], x] + g[x]*D[w[x, y], y] == h[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^x\frac {h(K[2]) \left (\left (y^{k+1}-(k+1) \int _1^x\frac {g(K[1])}{f(K[1])}dK[1]+(k+1) \int _1^{K[2]}\frac {g(K[1])}{f(K[1])}dK[1]\right ){}^{\frac {1}{k+1}}\right ){}^{-k}}{f(K[2])}dK[2]+c_1\left (\frac {y^{k+1}}{k+1}-\int _1^x\frac {g(K[1])}{f(K[1])}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := y^k*f(x)*diff(w(x,y),x) +g(x)*diff(w(x,y),y) = h(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \int _{}^{x}\frac {\left (\left (y^{k +1}+\int \frac {\left (-k -1\right ) g \left (x \right )}{f \left (x \right )}d x +\int \frac {\left (k +1\right ) g \left (\textit {\_b} \right )}{f \left (\textit {\_b} \right )}d \textit {\_b} \right )^{\frac {1}{k +1}}\right )^{-k} h \left (\textit {\_b} \right )}{f \left (\textit {\_b} \right )}d \textit {\_b} +\textit {\_F1} \left (y \,y^{k}+\left (-k -1\right ) \left (\int \frac {g \left (x \right )}{f \left (x \right )}d x \right )\right )\]
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Added Feb. 11, 2019.
Problem Chapter 3.8.1.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ y^k f(x) w_x + (y^{k+1} g_1(x) + g_0(x)) w_y = y^{3 k +1} h_2(x) + y^{2 k+1} h_1(x) + y^k h_0(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = y^k*f[x]*D[w[x, y], x] + (y^(k + 1)*g1[x] + g0[x])*D[w[x, y], y] == y^(3*k + 1)*h2[x] + y^(2*k + 1)*h1[x] + y^k*h0[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^x\frac {\text {h1}(K[3]) \left (\left (\exp \left (-\left ((k+1) \left (\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]-\int _1^{K[3]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right )\right )\right ) \left (y^{k+1}-\exp \left ((k+1) \int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) (k+1) \int _1^x\frac {\exp \left (-\left ((k+1) \int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right )\right ) \text {g0}(K[2])}{f(K[2])}dK[2]+\exp \left ((k+1) \int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) (k+1) \int _1^{K[3]}\frac {\exp \left (-\left ((k+1) \int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right )\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right )\right ){}^{\frac {1}{k+1}}\right ){}^{k+1}+\text {h2}(K[3]) \left (\left (\exp \left (-\left ((k+1) \left (\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]-\int _1^{K[3]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right )\right )\right ) \left (y^{k+1}-\exp \left ((k+1) \int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) (k+1) \int _1^x\frac {\exp \left (-\left ((k+1) \int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right )\right ) \text {g0}(K[2])}{f(K[2])}dK[2]+\exp \left ((k+1) \int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) (k+1) \int _1^{K[3]}\frac {\exp \left (-\left ((k+1) \int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right )\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right )\right ){}^{\frac {1}{k+1}}\right ){}^{2 k+1}+\text {h0}(K[3])}{f(K[3])}dK[3]+c_1\left (y^{k+1} \exp \left (-\left ((k+1) \int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right )\right )-(k+1) \int _1^x\frac {\exp \left (-\left ((k+1) \int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right )\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right )\right \}\right \}\]
Maple ✓
restart; pde := y^k*f(x)*diff(w(x,y),x) +(y^(k+1)* g1(x) + g0(x))*diff(w(x,y),y) = y^(3*k +1)*h2(x) + y^(2*k+1)*h1(x) + y^k*h0(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \int _{}^{x}\frac {\left (\left (\left (y^{k +1} {\mathrm e}^{-\left (k +1\right ) \left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )}+\left (k +1\right ) \left (\int \frac {{\mathrm e}^{-\left (k +1\right ) \left (\int \frac {\mathit {g1} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f} \right )} \mathit {g0} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f} \right )+\left (-k -1\right ) \left (\int \frac {{\mathrm e}^{-\left (k +1\right ) \left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )} \mathit {g0} \left (x \right )}{f \left (x \right )}d x \right )\right )^{\frac {1}{k +1}} {\mathrm e}^{\int \frac {\mathit {g1} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f}}\right )^{k} \mathit {h1} \left (\textit {\_f} \right )+\left (\left (y^{k +1} {\mathrm e}^{-\left (k +1\right ) \left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )}+\left (k +1\right ) \left (\int \frac {{\mathrm e}^{-\left (k +1\right ) \left (\int \frac {\mathit {g1} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f} \right )} \mathit {g0} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f} \right )+\left (-k -1\right ) \left (\int \frac {{\mathrm e}^{-\left (k +1\right ) \left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )} \mathit {g0} \left (x \right )}{f \left (x \right )}d x \right )\right )^{\frac {1}{k +1}} {\mathrm e}^{\int \frac {\mathit {g1} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f}}\right )^{2 k} \mathit {h2} \left (\textit {\_f} \right )\right ) \left (y^{k +1} {\mathrm e}^{-\left (k +1\right ) \left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )}+\left (k +1\right ) \left (\int \frac {{\mathrm e}^{-\left (k +1\right ) \left (\int \frac {\mathit {g1} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f} \right )} \mathit {g0} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f} \right )+\left (-k -1\right ) \left (\int \frac {{\mathrm e}^{-\left (k +1\right ) \left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )} \mathit {g0} \left (x \right )}{f \left (x \right )}d x \right )\right )^{\frac {1}{k +1}} {\mathrm e}^{\int \frac {\mathit {g1} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f}}+\mathit {h0} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f} +\textit {\_F1} \left (y^{k +1} {\mathrm e}^{-\left (k +1\right ) \left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )}+\left (-k -1\right ) \left (\int \frac {{\mathrm e}^{-\left (k +1\right ) \left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )} \mathit {g0} \left (x \right )}{f \left (x \right )}d x \right )\right )\]
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Added Feb. 11, 2019.
Problem Chapter 3.8.1.16 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ f(x) e^{\lambda x} w_x + g(x) w_y = h(x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = f[x]*Exp[lambda*x]*D[w[x, y], x] + g[x]*D[w[x, y], y] == h[x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^x\frac {e^{-\lambda K[2]} h(K[2])}{f(K[2])}dK[2]+c_1\left (y-\int _1^x\frac {e^{-\lambda K[1]} g(K[1])}{f(K[1])}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := f(x)*exp(lambda*x)*diff(w(x,y),x) +g(x)*diff(w(x,y),y) = h(x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \int \frac {{\mathrm e}^{-\lambda x} h \left (x \right )}{f \left (x \right )}d x +\textit {\_F1} \left (y -\left (\int \frac {{\mathrm e}^{-\lambda x} g \left (x \right )}{f \left (x \right )}d x \right )\right )\]
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