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 = 0.0365667 (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 = 0.022 (sec), leaf count = 50

\[ \left \{ y \relax (x ) ={\frac {1}{{x}^{2}} \left ({\frac {x}{a}{{\rm e}^{-{\frac {a{x}^{2}}{2}}}}}-{\frac {\sqrt {\pi }\sqrt {2}}{2}{\it Erf} \left ({\frac {\sqrt {2}x}{2}\sqrt {a}} \right ) {a}^{-{\frac {3}{2}}}}+{\it \_C1} \right ) {{\rm e}^{{\frac {a{x}^{2}}{2}}}}} \right \} \] 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),'implicit')

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