63.13.3 problem 4

Internal problem ID [13174]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.4.3 Reduction of order. Exercises page 125
Problem number : 4
Date solved : Tuesday, January 28, 2025 at 05:11:59 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }-2 a x^{\prime }+a^{2} x&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} x&={\mathrm e}^{a t} \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 14

dsolve([diff(x(t),t$2)-2*a*diff(x(t),t)+a^2*x(t)=0,exp(a*t)],singsol=all)
 

\[ x \left (t \right ) = {\mathrm e}^{a t} \left (c_{2} t +c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.020 (sec). Leaf size: 18

DSolve[D[x[t],{t,2}]-2*a*D[x[t],t]+a^2*x[t]==0,x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to e^{a t} (c_2 t+c_1) \]