2.12.14 problem 14

Internal problem ID [921]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.6, Forced Oscillations and Resonance. Page 362
Problem number : 14
Date solved : Tuesday, September 30, 2025 at 04:19:37 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+8 x^{\prime }+25 x&=200 \cos \left (t \right )+520 \sin \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=-30 \\ x^{\prime }\left (0\right )&=-10 \\ \end{align*}

✓ Maple. Time used: 0.045 (sec). Leaf size: 29

ode:=diff(diff(x(t),t),t)+8*diff(x(t),t)+25*x(t) = 200*cos(t)+520*sin(t); 
ic:=[x(0) = -30, D(x)(0) = -10]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \left (-31 \cos \left (3 t \right )-52 \sin \left (3 t \right )\right ) {\mathrm e}^{-4 t}+\cos \left (t \right )+22 \sin \left (t \right ) \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 34

ode=D[x[t],{t,2}]+8*D[x[t],t]+25*x[t]==200*Cos[t]+520*Sin[t]; 
ic={x[0]==-30,Derivative[1][x][0 ]==-10}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to 22 \sin (t)-52 e^{-4 t} \sin (3 t)+\cos (t)-31 e^{-4 t} \cos (3 t) \end{align*}

✓ Sympy. Time used: 0.159 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(25*x(t) - 520*sin(t) - 200*cos(t) + 8*Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): -30, Subs(Derivative(x(t), t), t, 0): -10} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (- 52 \sin {\left (3 t \right )} - 31 \cos {\left (3 t \right )}\right ) e^{- 4 t} + 22 \sin {\left (t \right )} + \cos {\left (t \right )} \]