13.9.7 problem 10

Internal problem ID [2393]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.2.2, Equal roots, reduction of order. Page 147
Problem number : 10
Date solved : Sunday, March 30, 2025 at 12:00:10 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: 0.218 (sec). Leaf size: 64

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]
 

\[ y(t)\to \frac {\sqrt {t^2+2 t-1} \left (c_1 \left (t^2-2 \left (\sqrt {2}-1\right ) t-2 \sqrt {2}+3\right )+c_2 (t+1)\right )}{\sqrt {-t^2-2 t+1}} \]

✓ Sympy. Time used: 0.972 (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)
 

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