14.9.12 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 : Monday, January 27, 2025 at 06:01:10 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

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

✓ Solution by Maple

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

dsolve([(2*t+1)*diff(y(t),t$2)-4*(t+1)*diff(y(t),t)+4*y(t)=0,1+t],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.051 (sec). Leaf size: 23

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

\[ y(t)\to c_1 e^{2 t+1}-c_2 (t+1) \]