8.12.3 problem 3

Internal problem ID [910]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.6, Forced Oscillations and Resonance. Page 362
Problem number : 3
Date solved : Tuesday, March 04, 2025 at 12:00:38 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+100 x&=225 \cos \left (5 t \right )+300 \sin \left (5 t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=375\\ x^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 29

ode:=diff(diff(x(t),t),t)+100*x(t) = 225*cos(5*t)+300*sin(5*t); 
ic:=x(0) = 375, D(x)(0) = 0; 
dsolve([ode,ic],x(t), singsol=all);
 

\[ x \left (t \right ) = -2 \sin \left (10 t \right )+372 \cos \left (10 t \right )+3 \cos \left (5 t \right )+4 \sin \left (5 t \right ) \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 30

ode=D[x[t],{t,2}]+100*x[t]==225*Cos[5*t]+300*Sin[5*t]; 
ic={x[0]==375,Derivative[1][x][0 ]==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to 4 \sin (5 t)-2 \sin (10 t)+3 \cos (5 t)+372 \cos (10 t) \]

✓ Sympy. Time used: 0.096 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(100*x(t) - 300*sin(5*t) - 225*cos(5*t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 375, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = 4 \sin {\left (5 t \right )} - 2 \sin {\left (10 t \right )} + 3 \cos {\left (5 t \right )} + 372 \cos {\left (10 t \right )} \]