57.10.3 problem 7(c)

Internal problem ID [14434]
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 2.3.2 Resonance Exercises page 114
Problem number : 7(c)
Date solved : Thursday, October 02, 2025 at 09:37:09 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+3025 x&=\cos \left (45 t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 17

ode:=diff(diff(x(t),t),t)+3025*x(t) = cos(45*t); 
ic:=[x(0) = 0, D(x)(0) = 0]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = -\frac {\cos \left (55 t \right )}{1000}+\frac {\cos \left (45 t \right )}{1000} \]

✓ Mathematica. Time used: 0.119 (sec). Leaf size: 36

ode=D[x[t],{t,2}]+55^2*x[t]==Cos[45*t]; 
ic={x[0]==0,Derivative[1][x][0 ]==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{250} \sin ^2(5 t) (\cos (5 t)+\cos (15 t)+\cos (25 t)+\cos (35 t)+\cos (45 t)) \end{align*}

✓ Sympy. Time used: 0.052 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(3025*x(t) - cos(45*t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {\cos {\left (45 t \right )}}{1000} - \frac {\cos {\left (55 t \right )}}{1000} \]