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 = 0.46637 (sec), leaf count = 152
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\exp \left (-\int _1^{K[3]}-f(K[1])dK[1]\right )}{\sqrt {c_1+2 \int _1^{K[3]}-\exp \left (-2 \int _1^{K[2]}-f(K[1])dK[1]\right ) g(K[2])dK[2]}}dK[3]\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[4]}-f(K[1])dK[1]\right )}{\sqrt {c_1+2 \int _1^{K[4]}-\exp \left (-2 \int _1^{K[2]}-f(K[1])dK[1]\right ) g(K[2])dK[2]}}dK[4]\& \right ][x+c_2]\right \}\right \}\]
Maple ✓
cpu = 0.623 (sec), leaf count = 104
\[\left [\int _{}^{y \left (x \right )}\frac {{\mathrm e}^{2 \left (\int f \left (\textit {\_f} \right )d \textit {\_f} \right )}}{\sqrt {-{\mathrm e}^{2 \left (\int f \left (\textit {\_f} \right )d \textit {\_f} \right )} \left (2 \left (\int {\mathrm e}^{\int 2 f \left (\textit {\_f} \right )d \textit {\_f}} g \left (\textit {\_f} \right )d \textit {\_f} \right )-\textit {\_C1} \right )}}d \textit {\_f} -x -\textit {\_C2} = 0, \int _{}^{y \left (x \right )}-\frac {{\mathrm e}^{2 \left (\int f \left (\textit {\_f} \right )d \textit {\_f} \right )}}{\sqrt {-{\mathrm e}^{2 \left (\int f \left (\textit {\_f} \right )d \textit {\_f} \right )} \left (2 \left (\int {\mathrm e}^{\int 2 f \left (\textit {\_f} \right )d \textit {\_f}} g \left (\textit {\_f} \right )d \textit {\_f} \right )-\textit {\_C1} \right )}}d \textit {\_f} -x -\textit {\_C2} = 0\right ]\] 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[Inactive[Integrate][-(1/(E^Inactive[Integrate][-f[K[1]], {K[1], 1, K[3]}]*Sqrt[C[1] + 2*Inactive[Integrate][-(g[K[2]]/E^(2*Inactive[Integrate][-f[K[1]], {K[1], 1, K[2]}])), {K[2], 1, K[3]}]])), {K[3], 1, #1}] & ][x + C[2]]}, {y[x] -> InverseFunction[Inactive[Integrate][1/(E^Inactive[Integrate][-f[K[1]], {K[1], 1, K[4]}]*Sqrt[C[1] + 2*Inactive[Integrate][-(g[K[2]]/E^(2*Inactive[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))
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]