4.37.16 $f(y(x))y^{\prime }(x{)}^2+g(y(x))+y^{″}(x)=0$

ODE
$$f(y(x))y^{\prime }(x{)}^2+g(y(x))+y^{″}(x)=0$$ ODE Classification

[[_2nd_order, _missing_x]]

Book solution method
TO DO

Mathematica ✓
cpu = 90.464 (sec), leaf count = 144

$$\{\{y(x)\to \text{InverseFunction}[{\int }_1^{\mathrm{\#}\text{1}}-\frac{e^{-{\int }_1^{K[3]}-f(K[1])dK[1]}}{\sqrt{2{\int }_1^{K[3]}g(K[2])(-e^{-2{\int }_1^{K[2]}-f(K[1])dK[1]})dK[2]+c_1}}dK[3]\mathrm{\&}][c_2+x]\},\{y(x)\to \text{InverseFunction}[{\int }_1^{\mathrm{\#}\text{1}}\frac{e^{-{\int }_1^{K[4]}-f(K[1])dK[1]}}{\sqrt{2{\int }_1^{K[4]}g(K[2])(-e^{-2{\int }_1^{K[2]}-f(K[1])dK[1]})dK[2]+c_1}}dK[4]\mathrm{\&}][c_2+x]\}\}$$

Maple ✓
cpu = 0.231 (sec), leaf count = 98

$$\{{\int }^{y\left(x\right)}{\mathrm{e}}^{2\int f\left(\mathit{\_}\mathit{f}\right)\mathrm{d}\mathit{\_}\mathit{f}}\frac{1}{\sqrt{{\mathrm{e}}^{2\int f\left(\mathit{\_}\mathit{f}\right)\mathrm{d}\mathit{\_}\mathit{f}}(-2\int {\left({\mathrm{e}}^{\int f\left(\mathit{\_}\mathit{f}\right)\mathrm{d}\mathit{\_}\mathit{f}}\right)}^{2}g\left(\mathit{\_}\mathit{f}\right)\mathrm{d}\mathit{\_}\mathit{f}+\mathit{\_}\mathit{C}\mathit{1})}}d\mathit{\_}\mathit{f}-x-\mathit{\_}\mathit{C}\mathit{2}=0,{\int }^{y\left(x\right)}-{\mathrm{e}}^{2\int f\left(\mathit{\_}\mathit{f}\right)\mathrm{d}\mathit{\_}\mathit{f}}\frac{1}{\sqrt{{\mathrm{e}}^{2\int f\left(\mathit{\_}\mathit{f}\right)\mathrm{d}\mathit{\_}\mathit{f}}(-2\int {\left({\mathrm{e}}^{\int f\left(\mathit{\_}\mathit{f}\right)\mathrm{d}\mathit{\_}\mathit{f}}\right)}^{2}g\left(\mathit{\_}\mathit{f}\right)\mathrm{d}\mathit{\_}\mathit{f}+\mathit{\_}\mathit{C}\mathit{1})}}d\mathit{\_}\mathit{f}-x-\mathit{\_}\mathit{C}\mathit{2}=0\}$$ 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[Inte
grate[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