2.12.4 problem 4

Internal problem ID [911]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.6, Forced Oscillations and Resonance. Page 362
Problem number : 4
Date solved : Tuesday, September 30, 2025 at 04:19:28 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+25 x&=90 \cos \left (4 t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ x^{\prime }\left (0\right )&=90 \\ \end{align*}

✓ Maple. Time used: 0.033 (sec). Leaf size: 23

ode:=diff(diff(x(t),t),t)+25*x(t) = 90*cos(4*t); 
ic:=[x(0) = 0, D(x)(0) = 90]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = 18 \sin \left (5 t \right )-10 \cos \left (5 t \right )+10 \cos \left (4 t \right ) \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 26

ode=D[x[t],{t,2}]+25*x[t]==90*Cos[4*t]; 
ic={x[0]==0,Derivative[1][x][0 ]==90}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to 2 (9 \sin (5 t)+5 \cos (4 t)-5 \cos (5 t)) \end{align*}

✓ Sympy. Time used: 0.048 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(25*x(t) - 90*cos(4*t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 90} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = 18 \sin {\left (5 t \right )} + 10 \cos {\left (4 t \right )} - 10 \cos {\left (5 t \right )} \]