59.1.612 problem 628

Internal problem ID [9784]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 628
Date solved : Wednesday, March 05, 2025 at 07:58:56 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

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

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

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

✓ Mathematica. Time used: 0.447 (sec). Leaf size: 98

ode=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; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.986 (sec). Leaf size: 19

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

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