4.12.48 \(y(x)^2 y'(x)=x \left (y(x)^2+1\right )\)

ODE
\[ y(x)^2 y'(x)=x \left (y(x)^2+1\right ) \] ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 1.40645 (sec), leaf count = 25

\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\text {$\#$1}-\tan ^{-1}(\text {$\#$1})\& \right ]\left [c_1+\frac {x^2}{2}\right ]\right \}\right \}\]

Maple
cpu = 0.008 (sec), leaf count = 17

\[ \left \{ {\frac {{x}^{2}}{2}}-y \relax (x ) +\arctan \left (y \relax (x ) \right ) +{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[y[x]^2*y'[x] == x*(1 + y[x]^2),y[x],x]

Mathematica raw output

{{y[x] -> InverseFunction[-ArcTan[#1] + #1 & ][x^2/2 + C[1]]}}

Maple raw input

dsolve(y(x)^2*diff(y(x),x) = x*(1+y(x)^2), y(x),'implicit')

Maple raw output

1/2*x^2-y(x)+arctan(y(x))+_C1 = 0