11.5.11 problem 12

Internal problem ID [1516]
Book : Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section : Chapter 6.5, The Laplace Transform. Impulse functions. page 273
Problem number : 12
Date solved : Monday, January 27, 2025 at 04:58:37 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\frac {\operatorname {Heaviside}\left (t -4+k \right )-\operatorname {Heaviside}\left (t -4-k \right )}{2 k} \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: 0.565 (sec). Leaf size: 83

dsolve([diff(y(t),t$2)+y(t)=1/(2*k)*(Heaviside(t-(4-k)) -  Heaviside(t-(4+k)) ),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \frac {\left (\operatorname {Heaviside}\left (4+k \right )+\operatorname {Heaviside}\left (t -4-k \right )-1\right ) \cos \left (-t +4+k \right )-\operatorname {Heaviside}\left (t -4-k \right )+\left (-\cos \left (t -4+k \right )+1\right ) \operatorname {Heaviside}\left (t -4+k \right )-\operatorname {Heaviside}\left (-4+k \right ) \cos \left (t \right )-\cos \left (t \right ) \operatorname {Heaviside}\left (4+k \right )+\operatorname {Heaviside}\left (-4+k \right ) \cos \left (t -4+k \right )+\cos \left (t \right )}{2 k} \]

✓ Solution by Mathematica

Time used: 1.081 (sec). Leaf size: 181

DSolve[{D[y[t],{t,2}]+y[t]==1/(2*k)*(UnitStep[t-(4-k)] -  UnitStep[t-(4+k)] ),{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} y(t)\to \fbox {$\frac {(\cos (k-t+4)-1) \theta (-k+t-4)-(\cos (-k-t+4)-1) \theta (k+t-4)}{2 k}\text { if }-4<k<4$} \\ y(t)\to \fbox {$\frac {\cos (-k-t+4)-\cos (t)+(\cos (k-t+4)-1) \theta (-k+t-4)-(\cos (-k-t+4)-1) \theta (k+t-4)}{2 k}\text { if }k>4$} \\ y(t)\to \fbox {$\frac {-\cos (k-t+4)+\cos (t)+(\cos (k-t+4)-1) \theta (-k+t-4)-(\cos (-k-t+4)-1) \theta (k+t-4)}{2 k}\text { if }k<-4$} \\ \end{align*}