59.1.382 problem 391

Internal problem ID [9554]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 391
Date solved : Monday, January 27, 2025 at 06:04:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (x^{2}+2\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 17

dsolve(x^2*diff(y(x), x, x) + 4*x*diff(y(x), x) + (x^2+2)*y(x) = 0,y(x), singsol=all)
 

\[ y = \frac {\sin \left (x \right ) c_{1} +\cos \left (x \right ) c_{2}}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.039 (sec). Leaf size: 37

DSolve[x^2*D[y[x],{x,2}]+4*x*D[y[x],x]+(x^2+2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {2 c_1 e^{-i x}-i c_2 e^{i x}}{2 x^2} \]