Internal problem ID [10962]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-4 Equation of form
\(x^2 y''+f(x)y'+g(x)y=0\)
Problem number: 128.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+2 y^{\prime } x -\left (a^{2} x^{2}+2\right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 34
dsolve(x^2*diff(y(x),x$2)+2*x*diff(y(x),x)-(a^2*x^2+2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{a x} \left (a x -1\right )}{x^{2}}+\frac {c_{2} {\mathrm e}^{-a x} \left (a x +1\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.042 (sec). Leaf size: 29
DSolve[x^2*y''[x]+2*x*y'[x]-(a^2*x^2+2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 j_{-2}(i a x)-c_2 y_{-2}(i a x) \]