59.1.18 problem 18

Internal problem ID [9190]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 18
Date solved : Monday, January 27, 2025 at 05:51:54 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 12

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

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

✓ Solution by Mathematica

Time used: 0.152 (sec). Leaf size: 90

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

\[ y(t)\to \exp \left (\int _1^t\frac {K[1]-2}{2 (K[1]-1)}dK[1]-\frac {1}{2} \int _1^t-\frac {K[2]}{K[2]-1}dK[2]\right ) \left (c_2 \int _1^t\exp \left (-2 \int _1^{K[3]}\frac {K[1]-2}{2 (K[1]-1)}dK[1]\right )dK[3]+c_1\right ) \]