59.1.350 problem 357

Internal problem ID [9522]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 357
Date solved : Monday, January 27, 2025 at 06:03:43 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.037 (sec). Leaf size: 21

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

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

✓ Solution by Mathematica

Time used: 0.035 (sec). Leaf size: 36

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

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