59.1.299 problem 302

Internal problem ID [9471]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 302
Date solved : Monday, January 27, 2025 at 06:03:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 20

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

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

✓ Solution by Mathematica

Time used: 0.033 (sec). Leaf size: 39

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

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