59.1.244 problem 247

Internal problem ID [9416]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 247
Date solved : Monday, January 27, 2025 at 06:02:35 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x^{2} y^{\prime }-2 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 21

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

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

✓ Solution by Mathematica

Time used: 0.068 (sec). Leaf size: 74

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

\[ y(x)\to -\frac {2 e^{\frac {x+1}{2}} \left ((c_1 x+2 i c_2) \cosh \left (\frac {x}{2}\right )-(i c_2 x+2 c_1) \sinh \left (\frac {x}{2}\right )\right )}{\sqrt {\pi } \sqrt {-i x} \sqrt {x}} \]