68.21.9 problem 24

Internal problem ID [17934]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 5. Applications of Higher Order Equations. Exercises 5.3, page 249
Problem number : 24
Date solved : Thursday, October 02, 2025 at 02:30:05 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+\frac {x^{\prime }}{10}+x&=3 \cos \left (2 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: 0.059 (sec). Leaf size: 46

ode:=diff(diff(x(t),t),t)+1/10*diff(x(t),t)+x(t) = 3*cos(2*t); 
ic:=[x(0) = 0, D(x)(0) = 0]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = -\frac {125 \,{\mathrm e}^{-\frac {t}{20}} \sqrt {399}\, \sin \left (\frac {\sqrt {399}\, t}{20}\right )}{30058}+\frac {225 \,{\mathrm e}^{-\frac {t}{20}} \cos \left (\frac {\sqrt {399}\, t}{20}\right )}{226}-\frac {225 \cos \left (2 t \right )}{226}+\frac {15 \sin \left (2 t \right )}{226} \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 74

ode=D[x[t],{t,2}]+1/10*D[x[t],t]+x[t]==3*Cos[2*t]; 
ic={x[0]==0,Derivative[1][x][0 ]==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -\frac {5 e^{-t/20} \left (-399 e^{t/20} \sin (2 t)+25 \sqrt {399} \sin \left (\frac {\sqrt {399} t}{20}\right )+5985 e^{t/20} \cos (2 t)-5985 \cos \left (\frac {\sqrt {399} t}{20}\right )\right )}{30058} \end{align*}

✓ Sympy. Time used: 0.193 (sec). Leaf size: 56

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(x(t) - 3*cos(2*t) + Derivative(x(t), t)/10 + 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)
 

\[ x{\left (t \right )} = \left (- \frac {125 \sqrt {399} \sin {\left (\frac {\sqrt {399} t}{20} \right )}}{30058} + \frac {225 \cos {\left (\frac {\sqrt {399} t}{20} \right )}}{226}\right ) e^{- \frac {t}{20}} + \frac {15 \sin {\left (2 t \right )}}{226} - \frac {225 \cos {\left (2 t \right )}}{226} \]