59.1.252 problem 255

Internal problem ID [9424]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 255
Date solved : Monday, January 27, 2025 at 06:02:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {9}{4}\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.048 (sec). Leaf size: 30

dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2-9/4)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {-c_{2} \left (-x +i\right ) {\mathrm e}^{-i x}+\left (x +i\right ) c_{1} {\mathrm e}^{i x}}{x^{{3}/{2}}} \]

✓ Solution by Mathematica

Time used: 0.060 (sec). Leaf size: 44

DSolve[x^2*D[y[x],{x,2}]+x*D[y[x],x]+(x^2-9/4)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\frac {\sqrt {\frac {2}{\pi }} ((c_1 x+c_2) \cos (x)+(c_2 x-c_1) \sin (x))}{x^{3/2}} \]