8.9.12 problem 15

Internal problem ID [2578]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.2.2. Equal roots, reduction of order. Excercises page 149
Problem number : 15
Date solved : Tuesday, September 30, 2025 at 05:47:23 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 t +1\right ) y^{\prime \prime }-4 \left (t +1\right ) y^{\prime }+4 y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=t +1 \end{align*}

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

ode:=(2*t+1)*diff(diff(y(t),t),t)-4*(t+1)*diff(y(t),t)+4*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 23

ode=(2*t+1)*D[y[t],{t,2}]-4*(1+t)*D[y[t],t]+4*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to c_1 e^{2 t+1}-c_2 (t+1) \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq((2*t + 1)*Derivative(y(t), (t, 2)) - (4*t + 4)*Derivative(y(t), t) + 4*y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

False