57.13.3 problem 4

Internal problem ID [14455]
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 : Thursday, October 02, 2025 at 09:37:28 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*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 14

ode:=diff(diff(x(t),t),t)-2*a*diff(x(t),t)+a^2*x(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

\[ x = {\mathrm e}^{a t} \left (c_2 t +c_1 \right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 18

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

\begin{align*} x(t)&\to e^{a t} (c_2 t+c_1) \end{align*}

✓ Sympy. Time used: 0.097 (sec). Leaf size: 12

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

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