4.8.7 \(x \left (1-x^2\right ) y'(x)=a x^2+y(x)\)

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

[_linear]

Book solution method
Linear ODE

Mathematica
cpu = 0.0169882 (sec), leaf count = 25

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

Maple
cpu = 0.024 (sec), leaf count = 51

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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