59.1.642 problem 659

Internal problem ID [9814]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 659
Date solved : Monday, January 27, 2025 at 06:14:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}-1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.040 (sec). Leaf size: 17

dsolve(4*x^2*diff(y(x),x$2)+4*x*diff(y(x),x)+(4*x^2-1)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.034 (sec). Leaf size: 39

DSolve[4*x^2*D[y[x],{x,2}]+4*x*D[y[x],x]+(4*x^2-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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