63.15.11 problem 11

Internal problem ID [13194]
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 : Tuesday, January 28, 2025 at 05:12:13 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*}

✓ Solution by Maple

Time used: 10.383 (sec). Leaf size: 26

dsolve([diff(x(t),t$2)+4*x(t)=cos(2*t)*Heaviside(2*Pi-t),x(0) = 0, D(x)(0) = 0],x(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 86.057 (sec). Leaf size: 129

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

\[ x(t)\to \sin (2 t) \left (\int _1^t\frac {1}{2} \cos ^2(2 K[2]) \theta (2 \pi -K[2])dK[2]-\int _1^0\frac {1}{2} \cos ^2(2 K[2]) \theta (2 \pi -K[2])dK[2]\right )-\cos (2 t) \int _1^0-\frac {1}{4} \sin (4 K[1]) \theta (2 \pi -K[1])dK[1]+\cos (2 t) \int _1^t-\frac {1}{4} \sin (4 K[1]) \theta (2 \pi -K[1])dK[1] \]