4.8.13 \(x \left (1-x^2\right ) y'(x)=\left (1-2 x^2\right ) y(x)+\left (1-x^2\right ) x^3\)

ODE
\[ x \left (1-x^2\right ) y'(x)=\left (1-2 x^2\right ) y(x)+\left (1-x^2\right ) x^3 \] ODE Classification

[_linear]

Book solution method
Linear ODE

Mathematica
cpu = 0.0128339 (sec), leaf count = 26

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

Maple
cpu = 0.011 (sec), leaf count = 23

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

DSolve[x*(1 - x^2)*y'[x] == x^3*(1 - x^2) + (1 - 2*x^2)*y[x],y[x],x]

Mathematica raw output

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

Maple raw input

dsolve(x*(-x^2+1)*diff(y(x),x) = x^3*(-x^2+1)+(-2*x^2+1)*y(x), y(x),'implicit')

Maple raw output

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