63.15.8 problem 6(h)

Internal problem ID [13191]
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 : 6(h)
Date solved : Tuesday, January 28, 2025 at 05:12:11 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+9 x&=\sin \left (3 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: 7.551 (sec). Leaf size: 18

dsolve([diff(x(t),t$2)+9*x(t)=sin(3*t),x(0) = 0, D(x)(0) = 0],x(t), singsol=all)
 

\[ x \left (t \right ) = \frac {\sin \left (3 t \right )}{18}-\frac {\cos \left (3 t \right ) t}{6} \]

✓ Solution by Mathematica

Time used: 0.061 (sec). Leaf size: 93

DSolve[{D[x[t],{t,2}]+9*x[t]==Sin[3*t],{x[0]==0,Derivative[1][x][0 ]==0}},x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \sin (3 t) \left (\int _1^t\frac {1}{6} \sin (6 K[2])dK[2]-\int _1^0\frac {1}{6} \sin (6 K[2])dK[2]\right )-\cos (3 t) \int _1^0-\frac {1}{3} \sin ^2(3 K[1])dK[1]+\cos (3 t) \int _1^t-\frac {1}{3} \sin ^2(3 K[1])dK[1] \]