4.3.39 \(y'(x)=f(x) g(y(x))\)

ODE
\[ y'(x)=f(x) g(y(x)) \] ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica ✓
cpu = 0.0899802 (sec), leaf count = 31

\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}} \frac {1}{g(K[1])} \, dK[1]\& \right ]\left [\int _1^x f(K[2]) \, dK[2]+c_1\right ]\right \}\right \}\]

Maple ✓
cpu = 0.007 (sec), leaf count = 20

\[ \left \{ \int \!f \left ( x \right ) \,{\rm d}x-\int ^{y \left ( x \right ) }\! \left ( g \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}+{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[y'[x] == f[x]*g[y[x]],y[x],x]

Mathematica raw output

{{y[x] -> InverseFunction[Integrate[g[K[1]]^(-1), {K[1], 1, #1}] & ][C[1] + Inte
grate[f[K[2]], {K[2], 1, x}]]}}

Maple raw input

dsolve(diff(y(x),x) = f(x)*g(y(x)), y(x),'implicit')

Maple raw output

Int(f(x),x)-Intat(1/g(_a),_a = y(x))+_C1 = 0