4.41.47 \(f(y(x)) y''(x)=y'(x)^2 f'(y(x))-g(x) f(y(x)) y'(x)-h(x) f(y(x))^2\)

ODE
\[ f(y(x)) y''(x)=y'(x)^2 f'(y(x))-g(x) f(y(x)) y'(x)-h(x) f(y(x))^2 \] ODE Classification

(ODEtools/info) missing specification of intermediate function

Book solution method
TO DO

Mathematica ✓
cpu = 10.406 (sec), leaf count = 70

\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}} \frac {1}{f(K[4])} \, dK[4]\& \right ]\left [\int _1^x -e^{-\int _1^{K[5]} g(K[1]) \, dK[1]} \left (\int _1^{K[5]} h(K[3]) e^{\int _1^{K[3]} g(K[1]) \, dK[1]} \, dK[3]+c_1\right ) \, dK[5]+c_2\right ]\right \}\right \}\]

Maple ✗
cpu = 0.059 (sec), leaf count = 0 , exception

unable to handle composite functions containing y(x) or diff(y(x),x) as in eval(diff(f(u),u),{u = y(x)})

Mathematica raw input

DSolve[f[y[x]]*y''[x] == -(f[y[x]]^2*h[x]) - f[y[x]]*g[x]*y'[x] + f'[y[x]]*y'[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> InverseFunction[Integrate[f[K[4]]^(-1), {K[4], 1, #1}] & ][C[2] + Inte
grate[-((C[1] + Integrate[E^Integrate[g[K[1]], {K[1], 1, K[3]}]*h[K[3]], {K[3], 
1, K[5]}])/E^Integrate[g[K[1]], {K[1], 1, K[5]}]), {K[5], 1, x}]]}}

Maple raw input

dsolve(f(y(x))*diff(diff(y(x),x),x) = eval(diff(f(u),u),{u = y(x)})*diff(y(x),x)^2-g(x)*f(y(x))*diff(y(x),x)-h(x)*f(y(x))^2, y(x),'implicit')

Maple raw output

unable to handle composite functions containing y(x) or diff(y(x),x) as in eval(
diff(f(u),u),{u = y(x)})