57.15.11 problem 11

Internal problem ID [14475]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 3, Laplace transform. Section 3.2.1 Initial value problems. Exercises page 156
Problem number : 11
Date solved : Thursday, October 02, 2025 at 09:37:38 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+4 x&=\cos \left (2 t \right ) \operatorname {Heaviside}\left (2 \pi -t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.354 (sec). Leaf size: 26

ode:=diff(diff(x(t),t),t)+4*x(t) = cos(2*t)*Heaviside(2*Pi-t); 
ic:=[x(0) = 0, D(x)(0) = 0]; 
dsolve([ode,op(ic)],x(t),method='laplace');
 

\[ x = -\frac {\left (-t +\operatorname {Heaviside}\left (-2 \pi +t \right ) \left (-2 \pi +t \right )\right ) \sin \left (2 t \right )}{4} \]

✓ Mathematica. Time used: 0.067 (sec). Leaf size: 28

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

\begin{align*} x(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} \pi \cos (t) \sin (t) & t>2 \pi \\ \frac {1}{2} t \cos (t) \sin (t) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.697 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(4*x(t) - cos(2*t)*Heaviside(-t + 2*pi) + 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)
 

\[ x{\left (t \right )} = \left (\frac {t \theta \left (- t + 2 \pi \right )}{4} - \frac {\pi \theta \left (- t + 2 \pi \right )}{2} + \frac {\pi }{2}\right ) \sin {\left (2 t \right )} \]