59.1.192 problem 194

Internal problem ID [9364]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 194
Date solved : Monday, January 27, 2025 at 06:01:56 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 15

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

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

✓ Solution by Mathematica

Time used: 0.449 (sec). Leaf size: 98

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

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