59.1.197 problem 199

Internal problem ID [9369]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 199
Date solved : Monday, January 27, 2025 at 06:01:59 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 t +1\right ) y^{\prime \prime }-4 \left (1+t \right ) y^{\prime }+4 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 15

dsolve((2*t+1)*diff(y(t),t$2)-4*(t+1)*diff(y(t),t)+4*y(t)=0,y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.182 (sec). Leaf size: 88

DSolve[(2*t+1)*D[y[t],{t,2}]-4*(t+1)*D[y[t],t]+4*y[t]==0,y[t],t,IncludeSingularSolutions -> True]
 

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