59.1.398 problem 410

Internal problem ID [9570]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 410
Date solved : Monday, January 27, 2025 at 06:04:13 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 1.592 (sec). Leaf size: 17

dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2-1/4)*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: 0.020 (sec). Leaf size: 33

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

\[ y(x)\to c_1 x-\frac {1}{2} c_2 (x \log (1-x)-x \log (x+1)+2) \]