61.29.19 problem 128

Internal problem ID [12628]
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
Problem number : 128
Date solved : Tuesday, January 28, 2025 at 03:23:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+2 x y^{\prime }-\left (a^{2} x^{2}+2\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 32

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)
 

\[ y = \frac {c_{1} {\mathrm e}^{a x} \left (a x -1\right )+c_{2} {\mathrm e}^{-a x} \left (a x +1\right )}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.032 (sec). Leaf size: 29

DSolve[x^2*D[y[x],{x,2}]+2*x*D[y[x],x]-(a^2*x^2+2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 j_{-2}(i a x)-c_2 y_{-2}(i a x) \]