ODE
\[ -x (a x+5) y'(x)+(3 a x+5) y(x)+x^2 y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0902348 (sec), leaf count = 41
\[\left \{\left \{y(x)\to \frac {c_2 x \left (a^2 x^2+4 a x+6\right )}{a^4}-c_1 x e^{a x} (a x-3)\right \}\right \}\]
Maple ✓
cpu = 0.043 (sec), leaf count = 33
\[ \left \{ y \left ( x \right ) = \left ( {\it \_C2}\, \left ( ax-3 \right ) {{\rm e}^{ax}}+{\it \_C1}\, \left ( {a}^{2}{x}^{2}+4\,ax+6 \right ) \right ) x \right \} \] Mathematica raw input
DSolve[(5 + 3*a*x)*y[x] - x*(5 + a*x)*y'[x] + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> -(E^(a*x)*x*(-3 + a*x)*C[1]) + (x*(6 + 4*a*x + a^2*x^2)*C[2])/a^4}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)-x*(a*x+5)*diff(y(x),x)+(3*a*x+5)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (_C2*(a*x-3)*exp(a*x)+_C1*(a^2*x^2+4*a*x+6))*x