Added March 10, 2019.
Problem Chapter 4.8.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = f(x) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == f[x]*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\int _1^x\frac {f(K[1])}{a}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) =f(x)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\int \frac {f \left (x \right )}{a}d x}\]
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Added March 10, 2019.
Problem Chapter 4.8.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + a w_y = f(x) y w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + a*D[w[x, y], y] == f[x]*y*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1(y-a x) \exp \left (\int _1^xf(K[1]) (y+a (K[1]-x))dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ a*diff(w(x,y),y) =f(x)*y*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (-a x +y \right ) {\mathrm e}^{\int _{}^{x}-\left (\left (-\textit {\_a} +x \right ) a -y \right ) f \left (\textit {\_a} \right )d \textit {\_a}}\]
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Added March 10, 2019.
Problem Chapter 4.8.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + a w_y = (f(x) y^2+g(x) y+h(x)) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + a*D[w[x, y], y] == (f[x]*y^2 + g[x]*y + h[x])*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1(y-a x) \exp \left (\int _1^x\left (f(K[1]) (y+a (K[1]-x))^2+g(K[1]) (y+a (K[1]-x))+h(K[1])\right )dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ a*diff(w(x,y),y) =(f(x)*y^2+g(x)*y+h(x))*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (-a x +y \right ) {\mathrm e}^{\int _{}^{x}\left (\left (\left (-\textit {\_a} +x \right ) a -y \right )^{2} f \left (\textit {\_a} \right )+\left (\left (\textit {\_a} -x \right ) a +y \right ) g \left (\textit {\_a} \right )+h \left (\textit {\_a} \right )\right )d \textit {\_a}}\]
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Added March 10, 2019.
Problem Chapter 4.8.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + a w_y = f(x) y^k w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + a*D[w[x, y], y] == f[x]*y^k*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1(y-a x) \exp \left (\int _1^xf(K[1]) (y+a (K[1]-x))^kdK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ a*diff(w(x,y),y) =f(x)*y^k*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (-a x +y \right ) {\mathrm e}^{\int _{}^{x}\left (\left (\textit {\_a} -x \right ) a +y \right )^{k} f \left (\textit {\_a} \right )d \textit {\_a}}\]
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Added March 10, 2019.
Problem Chapter 4.8.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + a w_y = f(x) e^{\lambda y} w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + a*D[w[x, y], y] == f[x]*Exp[lambda*y]*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1(y-a x) \exp \left (\int _1^xe^{\lambda (y+a (K[1]-x))} f(K[1])dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ a*diff(w(x,y),y) =f(x)*exp(lambda*y)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (-a x +y \right ) {\mathrm e}^{\int _{}^{x}{\mathrm e}^{-\left (\left (-\textit {\_a} +x \right ) a -y \right ) \lambda } f \left (\textit {\_a} \right )d \textit {\_a}}\]
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Added March 10, 2019.
Problem Chapter 4.8.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + (a y + f(x)) w_y = g(x) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*y + f[x])*D[w[x, y], y] == g[x]*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^xg(K[2])dK[2]\right ) c_1\left (y e^{-a x}-\int _1^xe^{-a K[1]} f(K[1])dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*y+f(x))*diff(w(x,y),y) =g(x)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (y \,{\mathrm e}^{-a x}-\left (\int {\mathrm e}^{-a x} f \left (x \right )d x \right )\right ) {\mathrm e}^{\int g \left (x \right )d x}\]
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Added March 10, 2019.
Problem Chapter 4.8.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + (a y + f(x)) w_y = g(x) y^k w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*y + f[x])*D[w[x, y], y] == g[x]*y^k*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y e^{-a x}-\int _1^xe^{-a K[1]} f(K[1])dK[1]\right ) \exp \left (\int _1^xg(K[2]) \left (e^{a K[2]} \left (e^{-a x} y-\int _1^xe^{-a K[1]} f(K[1])dK[1]+\int _1^{K[2]}e^{-a K[1]} f(K[1])dK[1]\right )\right ){}^kdK[2]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*y+f(x))*diff(w(x,y),y) =g(x)*y^k*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (y \,{\mathrm e}^{-a x}-\left (\int {\mathrm e}^{-a x} f \left (x \right )d x \right )\right ) {\mathrm e}^{\int _{}^{x}\left (\left (y \,{\mathrm e}^{-a x}+\int {\mathrm e}^{-\textit {\_b} a} f \left (\textit {\_b} \right )d \textit {\_b} -\left (\int {\mathrm e}^{-a x} f \left (x \right )d x \right )\right ) {\mathrm e}^{\textit {\_b} a}\right )^{k} g \left (\textit {\_b} \right )d \textit {\_b}}\]
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Added March 10, 2019.
Problem Chapter 4.8.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f(x) w_x + y^k w_y = g(x) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = f[x]*D[w[x, y], x] + y^k*D[w[x, y], y] == g[x]*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {g(K[2])}{f(K[2])}dK[2]\right ) c_1\left (-\int _1^x\frac {1}{f(K[1])}dK[1]-\frac {y^{1-k}}{k-1}\right )\right \}\right \}\]
Maple ✓
restart; pde := f(x)*diff(w(x,y),x)+ y^k*diff(w(x,y),y) =g(x)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\left (\left (k -1\right ) y^{k} \left (\int \frac {1}{f \left (x \right )}d x \right )+y \right ) y^{-k}\right ) {\mathrm e}^{\int \frac {g \left (x \right )}{f \left (x \right )}d x}\]
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Added March 10, 2019.
Problem Chapter 4.8.1.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f(x) w_x + (y+a) w_y = (b y+c) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = f[x]*D[w[x, y], x] + (y + a)*D[w[x, y], y] == (b*y + c)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y \exp \left (-\int _1^x\frac {1}{f(K[1])}dK[1]\right )-\int _1^x\frac {a \exp \left (-\int _1^{K[2]}\frac {1}{f(K[1])}dK[1]\right )}{f(K[2])}dK[2]\right ) \exp \left (\int _1^x\frac {c+b \exp \left (\int _1^{K[3]}\frac {1}{f(K[1])}dK[1]\right ) \left (\exp \left (-\int _1^x\frac {1}{f(K[1])}dK[1]\right ) y-\int _1^x\frac {a \exp \left (-\int _1^{K[2]}\frac {1}{f(K[1])}dK[1]\right )}{f(K[2])}dK[2]+\int _1^{K[3]}\frac {a \exp \left (-\int _1^{K[2]}\frac {1}{f(K[1])}dK[1]\right )}{f(K[2])}dK[2]\right )}{f(K[3])}dK[3]\right )\right \}\right \}\]
Maple ✓
restart; pde := f(x)*diff(w(x,y),x)+ (y+a)*diff(w(x,y),y) =(b*y+c)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\left (a +y \right ) {\mathrm e}^{-\left (\int \frac {1}{f \left (x \right )}d x \right )}\right ) {\mathrm e}^{\left (a +y \right ) b +\left (-a b +c \right ) \left (\int \frac {1}{f \left (x \right )}d x \right )}\]
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Added March 10, 2019.
Problem Chapter 4.8.1.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f(x) w_x + (y+a x) w_y = g(x) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = f[x]*D[w[x, y], x] + (y + a*x)*D[w[x, y], y] == g[x]*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {g(K[3])}{f(K[3])}dK[3]\right ) c_1\left (y \exp \left (-\int _1^x\frac {1}{f(K[1])}dK[1]\right )-\int _1^x\frac {a \exp \left (-\int _1^{K[2]}\frac {1}{f(K[1])}dK[1]\right ) K[2]}{f(K[2])}dK[2]\right )\right \}\right \}\]
Maple ✓
restart; pde := f(x)*diff(w(x,y),x)+ (y+a*x)*diff(w(x,y),y) =g(x)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (-a \left (\int \frac {x \,{\mathrm e}^{-\left (\int \frac {1}{f \left (x \right )}d x \right )}}{f \left (x \right )}d x \right )+y \,{\mathrm e}^{-\left (\int \frac {1}{f \left (x \right )}d x \right )}\right ) {\mathrm e}^{\int \frac {g \left (x \right )}{f \left (x \right )}d x}\]
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Added March 10, 2019.
Problem Chapter 4.8.1.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f(x) w_x + (g_1(x) y+g_0(x)) w_y = \left ( h_2(x) y^2+ h_1(x)y + h_0(x) \right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = f[x]*D[w[x, y], x] + (g1[x]*y + g0[x])*D[w[x, y], y] == (h2[x]*y^2 + h1[x]*y + h0[x])*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y \exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right )-\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right ) \exp \left (\int _1^x\frac {\exp \left (2 \int _1^{K[3]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {h2}(K[3]) \left (\exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) y-\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]+\int _1^{K[3]}\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right ){}^2+\exp \left (\int _1^{K[3]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {h1}(K[3]) \left (\exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) y-\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]+\int _1^{K[3]}\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right )+\text {h0}(K[3])}{f(K[3])}dK[3]\right )\right \}\right \}\]
Maple ✓
restart; pde := f(x)*diff(w(x,y),x)+ (g1(x)*y+g0(x))*diff(w(x,y),y) =(h2(x)*y^2+h1(x)*y+h0(x))*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (y \,{\mathrm e}^{-\left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )}-\left (\int \frac {{\mathrm e}^{-\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 ) {\mathrm e}^{\int _{}^{x}\frac {\left (\int \frac {{\mathrm e}^{-\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 )^{2} {\mathrm e}^{2 \left (\int \frac {\mathit {g1} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f} \right )} \mathit {h2} \left (\textit {\_f} \right )+2 \left (y \,{\mathrm e}^{-\left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )}-\left (\int \frac {{\mathrm e}^{-\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 ) \left (\int \frac {{\mathrm e}^{-\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 ) {\mathrm e}^{2 \left (\int \frac {\mathit {g1} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f} \right )} \mathit {h2} \left (\textit {\_f} \right )+\left (\int \frac {{\mathrm e}^{-\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 ) {\mathrm e}^{\int \frac {\mathit {g1} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f}} \mathit {h1} \left (\textit {\_f} \right )+\left (y \,{\mathrm e}^{-\left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )}-\left (\int \frac {{\mathrm e}^{-\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 )^{2} {\mathrm e}^{2 \left (\int \frac {\mathit {g1} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f} \right )} \mathit {h2} \left (\textit {\_f} \right )+\left (y \,{\mathrm e}^{-\left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )}-\left (\int \frac {{\mathrm e}^{-\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 ) {\mathrm e}^{\int \frac {\mathit {g1} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f}} \mathit {h1} \left (\textit {\_f} \right )+\mathit {h0} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f}}\]
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Added March 10, 2019.
Problem Chapter 4.8.1.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f(x) w_x + (g_1(x) y+g_2(x) y^k) w_y = h(x) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = f[x]*D[w[x, y], x] + (g1[x]*y + g2[x]*y^k)*D[w[x, y], y] == h[x]*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {h(K[3])}{f(K[3])}dK[3]\right ) c_1\left ((k-1) \int _1^x\frac {\exp \left ((k-1) \int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g2}(K[2])}{f(K[2])}dK[2]+y^{1-k} \exp \left ((k-1) \int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := f(x)*diff(w(x,y),x)+ (g1(x)*y+g2(x)*y^k)*diff(w(x,y),y) =h(x)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); sol := simplify(sol);
\[w \left (x , y\right ) = \textit {\_F1} \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )}+\left (k -1\right ) \left (\int \frac {{\mathrm e}^{\left (k -1\right ) \left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )} \mathit {g2} \left (x \right )}{f \left (x \right )}d x \right )\right ) {\mathrm e}^{\int \frac {h \left (x \right )}{f \left (x \right )}d x}\]
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Added March 10, 2019.
Problem Chapter 4.8.1.13, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f(x) w_x + (g_1(x)+g_2(x) e^{\lambda y}) w_y = h(x) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = f[x]*D[w[x, y], x] + (g1[x]*y + g2[x]*Exp[lambda*y])*D[w[x, y], y] == h[x]*w[x, y]; 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)*y+g2(x)*exp(lambda*y))*diff(w(x,y),y) =h(x)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
sol=()
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Added March 10, 2019.
Problem Chapter 4.8.1.14, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f(x) y^k w_x + g(x) w_y = h(x) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = f[x]*y^k*D[w[x, y], x] + g[x]*D[w[x, y], y] == h[x]*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {y^{k+1}}{k+1}-\int _1^x\frac {g(K[1])}{f(K[1])}dK[1]\right ) \exp \left (\int _1^x\frac {h(K[2]) \left (\left (y^{k+1}-(k+1) \int _1^x\frac {g(K[1])}{f(K[1])}dK[1]+(k+1) \int _1^{K[2]}\frac {g(K[1])}{f(K[1])}dK[1]\right ){}^{\frac {1}{k+1}}\right ){}^{-k}}{f(K[2])}dK[2]\right )\right \}\right \}\]
Maple ✓
restart; pde := f(x)*y^k*diff(w(x,y),x)+ g(x)*diff(w(x,y),y) =h(x)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (y \,y^{k}+\left (-k -1\right ) \left (\int \frac {g \left (x \right )}{f \left (x \right )}d x \right )\right ) {\mathrm e}^{\int _{}^{x}\frac {\left (\left (y^{k +1}+\int \frac {\left (-k -1\right ) g \left (x \right )}{f \left (x \right )}d x +\int \frac {\left (k +1\right ) g \left (\textit {\_b} \right )}{f \left (\textit {\_b} \right )}d \textit {\_b} \right )^{\frac {1}{k +1}}\right )^{-k} h \left (\textit {\_b} \right )}{f \left (\textit {\_b} \right )}d \textit {\_b}}\]
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Added March 10, 2019.
Problem Chapter 4.8.1.15, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f(x) e^{\lambda y} w_x + g(x) w_y = h(x) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = f[x]*Exp[lambda*y]*D[w[x, y], x] + g[x]*D[w[x, y], y] == h[x]*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{\lambda y}}{\lambda }-\int _1^x\frac {g(K[1])}{f(K[1])}dK[1]\right ) \exp \left (\int _1^x\frac {h(K[2])}{f(K[2]) \left (-\lambda \int _1^x\frac {g(K[1])}{f(K[1])}dK[1]+e^{\lambda y}+\lambda \int _1^{K[2]}\frac {g(K[1])}{f(K[1])}dK[1]\right )}dK[2]\right )\right \}\right \}\]
Maple ✓
restart; pde := f(x)*exp(lambda*y)*diff(w(x,y),x)+ g(x)*diff(w(x,y),y) =h(x)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {-\lambda \left (\int \frac {g \left (x \right )}{f \left (x \right )}d x \right )+{\mathrm e}^{\lambda y}}{\lambda }\right ) {\mathrm e}^{\int _{}^{x}\frac {h \left (\textit {\_b} \right )}{\left (\lambda \left (\int \frac {g \left (\textit {\_b} \right )}{f \left (\textit {\_b} \right )}d \textit {\_b} \right )-\lambda \left (\int \frac {g \left (x \right )}{f \left (x \right )}d x \right )+{\mathrm e}^{\lambda y}\right ) f \left (\textit {\_b} \right )}d \textit {\_b}}\]
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