59.1.14 problem 14

Internal problem ID [9186]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 14
Date solved : Monday, January 27, 2025 at 05:51:51 PM
CAS classification : [_Laguerre]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 13

dsolve(t*diff(y(t),t$2)-(1+t)*diff(y(t),t)+y(t) = 0,y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.422 (sec). Leaf size: 78

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

\[ y(t)\to \sqrt {t} \exp \left (\frac {1}{2} \left (2 \int _1^t\frac {K[1]-1}{2 K[1]}dK[1]+t+1\right )\right ) \left (c_2 \int _1^t\exp \left (-2 \int _1^{K[2]}\frac {K[1]-1}{2 K[1]}dK[1]\right )dK[2]+c_1\right ) \]