74.9.16 problem 27

Internal problem ID [16192]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.1, page 141
Problem number : 27
Date solved : Tuesday, January 28, 2025 at 08:56:54 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+4 y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{2 t} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 10

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

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

✓ Solution by Mathematica

Time used: 0.016 (sec). Leaf size: 12

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

\[ y(t)\to e^{2 t} t \]