59.1.619 problem 635

Internal problem ID [9791]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 635
Date solved : Sunday, March 30, 2025 at 02:46:43 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*}

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

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

\[ y = c_2 \,t^{2}+c_1 t -c_2 \]

✓ Mathematica. Time used: 0.308 (sec). Leaf size: 79

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

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

✓ Sympy. Time used: 0.841 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-2*t*Derivative(y(t), t)/(t**2 + 1) + Derivative(y(t), (t, 2)) + 2*y(t)/(t**2 + 1),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{2} \left (1 - t^{2}\right ) + C_{1} t + O\left (t^{6}\right ) \]