Internal problem ID [8682]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1102.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {y^{\prime \prime } x +2 y^{\prime }+a \,x^{2} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 35
dsolve(x*diff(diff(y(x),x),x)+2*diff(y(x),x)+a*x^2*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \BesselJ \left (\frac {1}{3}, \frac {2 \sqrt {a}\, x^{\frac {3}{2}}}{3}\right )}{\sqrt {x}}+\frac {c_{2} \BesselY \left (\frac {1}{3}, \frac {2 \sqrt {a}\, x^{\frac {3}{2}}}{3}\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 36
DSolve[a*x^2*y[x] + 2*y'[x] + x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1 \text {Ai}\left (\sqrt [3]{-a} x\right )+c_2 \text {Bi}\left (\sqrt [3]{-a} x\right )}{x} \\ \end{align*}