59.1.199 problem 201

Internal problem ID [9371]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 201
Date solved : Monday, January 27, 2025 at 06:02:01 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 16

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

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

✓ Solution by Mathematica

Time used: 0.302 (sec). Leaf size: 79

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

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