59.1.625 problem 642

Internal problem ID [9797]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 642
Date solved : Monday, January 27, 2025 at 06:14:23 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 17

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

\[ y = \frac {c_{2} {\mathrm e}^{t}+c_{1} t +c_{1}}{t} \]

✓ Solution by Mathematica

Time used: 0.249 (sec). Leaf size: 80

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

\begin{align*} y(t)\to \exp \left (\int _1^t\left (1-\frac {1}{K[1]}\right )dK[1]\right ) \left (\int _1^t\exp \left (-\int _1^{K[2]}\left (1-\frac {1}{K[1]}\right )dK[1]\right ) c_1dK[2]+c_2\right ) \\ y(t)\to c_2 \exp \left (\int _1^t\left (1-\frac {1}{K[1]}\right )dK[1]\right ) \\ \end{align*}