ODE
\[ f(y(x)) y'(x)^2+g(y(x))+y''(x)=0 \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 90.464 (sec), leaf count = 144
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}} -\frac {e^{-\int _1^{K[3]} -f(K[1]) \, dK[1]}}{\sqrt {2 \int _1^{K[3]} g(K[2]) \left (-e^{-2 \int _1^{K[2]} -f(K[1]) \, dK[1]}\right ) \, dK[2]+c_1}} \, dK[3]\& \right ]\left [c_2+x\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}} \frac {e^{-\int _1^{K[4]} -f(K[1]) \, dK[1]}}{\sqrt {2 \int _1^{K[4]} g(K[2]) \left (-e^{-2 \int _1^{K[2]} -f(K[1]) \, dK[1]}\right ) \, dK[2]+c_1}} \, dK[4]\& \right ]\left [c_2+x\right ]\right \}\right \}\]
Maple ✓
cpu = 0.231 (sec), leaf count = 98
\[ \left \{ \int ^{y \left ( x \right ) }\!{{{\rm e}^{2\,\int \!f \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}{\frac {1}{\sqrt {{{\rm e}^{2\,\int \!f \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}} \left ( -2\,\int \! \left ( {{\rm e}^{\int \!f \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}} \right ) ^{2}g \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}+{\it \_C1} \right ) }}}}{d{\it \_f}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!-{{{\rm e}^{2\,\int \!f \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}}{\frac {1}{\sqrt {{{\rm e}^{2\,\int \!f \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}} \left ( -2\,\int \! \left ( {{\rm e}^{\int \!f \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}}} \right ) ^{2}g \left ( {\it \_f} \right ) \,{\rm d}{\it \_f}+{\it \_C1} \right ) }}}}{d{\it \_f}}-x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[g[y[x]] + f[y[x]]*y'[x]^2 + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[Integrate[-(1/(E^Integrate[-f[K[1]], {K[1], 1, K[3]}]*Sqrt[C[1] + 2*Integrate[-(g[K[2]]/E^(2*Integrate[-f[K[1]], {K[1], 1, K[2]}])), {K[2], 1, K[3]}]])), {K[3], 1, #1}] & ][x + C[2]]}, {y[x] -> InverseFunction[Integrate[1/(E^Integrate[-f[K[1]], {K[1], 1, K[4]}]*Sqrt[C[1] + 2*Integrate[-(g[K[2]]/E^(2*Integrate[-f[K[1]], {K[1], 1, K[2]}])), {K[2], 1, K[4]}]]), {K[4], 1, #1}] & ][x + C[2]]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+f(y(x))*diff(y(x),x)^2+g(y(x)) = 0, y(x),'implicit')
Maple raw output
Intat(exp(2*Int(f(_f),_f))/(exp(2*Int(f(_f),_f))*(-2*Int(exp(Int(f(_f),_f))^2*g(_f),_f)+_C1))^(1/2),_f = y(x))-x-_C2 = 0, Intat(-exp(2*Int(f(_f),_f))/(exp(2*Int(f(_f),_f))*(-2*Int(exp(Int(f(_f),_f))^2*g(_f),_f)+_C1))^(1/2),_f = y(x))-x-_C2 = 0