4.14.18 \(x \left (x y(x)^2+1\right ) y'(x)+y(x)=0\)

ODE
\[ x \left (x y(x)^2+1\right ) y'(x)+y(x)=0 \] ODE Classification

[[_homogeneous, `class G`], _rational]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.0167009 (sec), leaf count = 60

\[\left \{\left \{y(x)\to \frac {1}{2} \left (c_1-\frac {\sqrt {c_1^2 x+4}}{\sqrt {x}}\right )\right \},\left \{y(x)\to \frac {1}{2} \left (\frac {\sqrt {c_1^2 x+4}}{\sqrt {x}}+c_1\right )\right \}\right \}\]

Maple
cpu = 0.024 (sec), leaf count = 40

\[ \left \{ \ln \relax (x ) -{\it \_C1}-2\,\ln \left (y \relax (x ) \sqrt {x}-1 \right ) -2\,\ln \left (y \relax (x ) \sqrt {x}+1 \right ) +2\,\ln \left (y \relax (x ) \sqrt {x} \right ) =0 \right \} \] Mathematica raw input

DSolve[y[x] + x*(1 + x*y[x]^2)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (C[1] - Sqrt[4 + x*C[1]^2]/Sqrt[x])/2}, {y[x] -> (C[1] + Sqrt[4 + x*C[
1]^2]/Sqrt[x])/2}}

Maple raw input

dsolve(x*(1+x*y(x)^2)*diff(y(x),x)+y(x) = 0, y(x),'implicit')

Maple raw output

ln(x)-_C1-2*ln(y(x)*x^(1/2)-1)-2*ln(y(x)*x^(1/2)+1)+2*ln(y(x)*x^(1/2)) = 0