ODE
\[ f\left (y'(x)\right )+x g\left (y'(x)\right )=y(x) \] ODE Classification
[_dAlembert]
Book solution method
Clairaut’s equation and related types, d’Alembert’s equation (also call Lagrange’s)
Mathematica ✓
cpu = 0.233448 (sec), leaf count = 87
\[\text {Solve}\left [\left \{x=\exp \left (\int _1^{\text {K$\$$271269}} \frac {g'(K[1])}{K[1]-g(K[1])} \, dK[1]\right ) \left (\int \frac {f'(\text {K$\$$271269}) \exp \left (-\int _1^{\text {K$\$$271269}} \frac {g'(K[1])}{K[1]-g(K[1])} \, dK[1]\right )}{\text {K$\$$271269}-g(\text {K$\$$271269})} \, d\text {K$\$$271269}+c_1\right ),f(\text {K$\$$271269})+x g(\text {K$\$$271269})=y(x)\right \},\{y(x),\text {K$\$$271269}\}\right ]\]
Maple ✓
cpu = 0.099 (sec), leaf count = 138
\[ \left \{ y \left ( x \right ) =x{\it RootOf} \left ( {\it \_Z}-g \left ( {\it \_Z} \right ) \right ) +f \left ( {\it RootOf} \left ( {\it \_Z}-g \left ( {\it \_Z} \right ) \right ) \right ) ,[x \left ( {\it \_T} \right ) ={{\rm e}^{\int \!{\frac {{\frac {\rm d}{{\rm d}{\it \_T}}}g \left ( {\it \_T} \right ) }{{\it \_T}-g \left ( {\it \_T} \right ) }}\,{\rm d}{\it \_T}}} \left ( \int \!{\frac {{\frac {\rm d}{{\rm d}{\it \_T}}}f \left ( {\it \_T} \right ) }{{\it \_T}-g \left ( {\it \_T} \right ) }{{\rm e}^{-\int \!{\frac {{\frac {\rm d}{{\rm d}{\it \_T}}}g \left ( {\it \_T} \right ) }{{\it \_T}-g \left ( {\it \_T} \right ) }}\,{\rm d}{\it \_T}}}}\,{\rm d}{\it \_T}+{\it \_C1} \right ) ,y \left ( {\it \_T} \right ) =f \left ( {\it \_T} \right ) +{{\rm e}^{\int \!{\frac {{\frac {\rm d}{{\rm d}{\it \_T}}}g \left ( {\it \_T} \right ) }{{\it \_T}-g \left ( {\it \_T} \right ) }}\,{\rm d}{\it \_T}}} \left ( \int \!{\frac {{\frac {\rm d}{{\rm d}{\it \_T}}}f \left ( {\it \_T} \right ) }{{\it \_T}-g \left ( {\it \_T} \right ) }{{\rm e}^{-\int \!{\frac {{\frac {\rm d}{{\rm d}{\it \_T}}}g \left ( {\it \_T} \right ) }{{\it \_T}-g \left ( {\it \_T} \right ) }}\,{\rm d}{\it \_T}}}}\,{\rm d}{\it \_T}+{\it \_C1} \right ) g \left ( {\it \_T} \right ) ] \right \} \] Mathematica raw input
DSolve[f[y'[x]] + x*g[y'[x]] == y[x],y[x],x]
Mathematica raw output
Solve[{x == E^Integrate[Derivative[1][g][K[1]]/(-g[K[1]] + K[1]), {K[1], 1, K$271269}]*(C[1] + Integrate[Derivative[1][f][K$271269]/(E^Integrate[Derivative[1][g][K[1]]/(-g[K[1]] + K[1]), {K[1], 1, K$271269}]*(K$271269 - g[K$271269])), K$271269]), f[K$271269] + x*g[K$271269] == y[x]}, {y[x], K$271269}]
Maple raw input
dsolve(f(diff(y(x),x))+x*g(diff(y(x),x)) = y(x), y(x),'implicit')
Maple raw output
y(x) = x*RootOf(_Z-g(_Z))+f(RootOf(_Z-g(_Z))), [x(_T) = exp(Int(1/(_T-g(_T))*diff(g(_T),_T),_T))*(Int(1/(_T-g(_T))*diff(f(_T),_T)*exp(-Int(1/(_T-g(_T))*diff(g(_T),_T),_T)),_T)+_C1), y(_T) = f(_T)+exp(Int(1/(_T-g(_T))*diff(g(_T),_T),_T))*(Int(1/(_T-g(_T))*diff(f(_T),_T)*exp(-Int(1/(_T-g(_T))*diff(g(_T),_T),_T)),_T)+_C1)*g(_T)]