##### 4.4.15 $$\left (2-a x^2\right ) y(x)+x y'(x)+x=0$$

ODE
$\left (2-a x^2\right ) y(x)+x y'(x)+x=0$ ODE Classiﬁcation

[_linear]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.183791 (sec), leaf count = 70

$\left \{\left \{y(x)\to \frac {-\frac {\sqrt {2 \pi } e^{\frac {a x^2}{2}} \text {erf}\left (\frac {\sqrt {a} x}{\sqrt {2}}\right )}{a^{3/2}}+2 c_1 e^{\frac {a x^2}{2}}+\frac {2 x}{a}}{2 x^2}\right \}\right \}$

Maple
cpu = 0.023 (sec), leaf count = 50

$\left [y \left (x \right ) = \frac {\left (\frac {{\mathrm e}^{-\frac {a \,x^{2}}{2}} x}{a}-\frac {\sqrt {\pi }\, \sqrt {2}\, \erf \left (\frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right )}{2 a^{\frac {3}{2}}}+\textit {\_C1} \right ) {\mathrm e}^{\frac {a \,x^{2}}{2}}}{x^{2}}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> ((2*x)/a + 2*E^((a*x^2)/2)*C[1] - (E^((a*x^2)/2)*Sqrt[2*Pi]*Erf[(Sqrt[
a]*x)/Sqrt[2]])/a^(3/2))/(2*x^2)}}

Maple raw input

dsolve(x*diff(y(x),x)+x+(-a*x^2+2)*y(x) = 0, y(x))

Maple raw output

[y(x) = (exp(-1/2*a*x^2)*x/a-1/2/a^(3/2)*Pi^(1/2)*2^(1/2)*erf(1/2*2^(1/2)*a^(1/2
)*x)+_C1)*exp(1/2*a*x^2)/x^2]