67.4.17 problem Problem 3(c)

Internal problem ID [14061]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 3(c)
Date solved : Tuesday, January 28, 2025 at 06:13:22 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+9 y&=24 \sin \left (t \right ) \left (\operatorname {Heaviside}\left (t \right )+\operatorname {Heaviside}\left (t -\pi \right )\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 9.366 (sec). Leaf size: 18

dsolve([diff(y(t),t$2)+9*y(t)=24*sin(t)*(Heaviside(t)+Heaviside(t-Pi)),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = 4 \left (1+\operatorname {Heaviside}\left (t -\pi \right )\right ) \sin \left (t \right )^{3} \]

✓ Solution by Mathematica

Time used: 0.045 (sec). Leaf size: 24

DSolve[{D[y[t],{t,2}]+9*y[t]==24*Sin[t]*(UnitStep[t]+UnitStep[t-Pi]),{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to 4 (\theta (\pi -t) (\theta (t)-2)+2) \sin ^3(t) \]