Internal problem ID [8860]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1280.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 y^{\prime \prime } x^{2}+8 x y^{\prime }-\left (4 x^{2}+12 x +3\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 40
dsolve(4*x^2*diff(diff(y(x),x),x)+8*x*diff(y(x),x)-(4*x^2+12*x+3)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sqrt {x}\, {\mathrm e}^{x}+\frac {c_{2} \left (-4 x^{2} {\mathrm e}^{x} \expIntegral \left (1, 2 x \right )+{\mathrm e}^{-x} \left (-1+2 x \right )\right )}{x^{\frac {3}{2}}} \]
✓ Solution by Mathematica
Time used: 0.035 (sec). Leaf size: 47
DSolve[(-3 - 12*x - 4*x^2)*y[x] + 8*x*y'[x] + 4*x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^x \sqrt {x} (2 c_2 \text {Ei}(-2 x)+c_1)+\frac {c_2 e^{-x} (2 x-1)}{2 x^{3/2}} \\ \end{align*}