63.10.3 problem 7(c)

Internal problem ID [13082]
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 : Monday, March 31, 2025 at 07:33:49 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,ic],x(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.171 (sec). Leaf size: 109

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]
 

\[ x(t)\to -\sin (55 t) \int _1^0\frac {1}{55} \cos (45 K[2]) \cos (55 K[2])dK[2]+\sin (55 t) \int _1^t\frac {1}{55} \cos (45 K[2]) \cos (55 K[2])dK[2]+\cos (55 t) \left (\int _1^t-\frac {1}{55} \cos (45 K[1]) \sin (55 K[1])dK[1]-\int _1^0-\frac {1}{55} \cos (45 K[1]) \sin (55 K[1])dK[1]\right ) \]

✓ Sympy. Time used: 0.103 (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)
 

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