5.5.11 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 : Saturday, October 04, 2025 at 04:13:53 PM
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*}

✓ Maple. Time used: 0.330 (sec). Leaf size: 83

ode:=diff(diff(y(t),t),t)+y(t) = 1/2/k*(Heaviside(t-4+k)-Heaviside(t-4-k)); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.919 (sec). Leaf size: 181

ode=D[y[t],{t,2}]+y[t]==1/(2*k)*(UnitStep[t-(4-k)] -  UnitStep[t-(4+k)] ); 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 0.769 (sec). Leaf size: 201

from sympy import * 
t = symbols("t") 
k = symbols("k") 
y = Function("y") 
ode = Eq(y(t) + Derivative(y(t), (t, 2)) - (-Heaviside(-k + t - 4) + Heaviside(k + t - 4))/(2*k),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\frac {\cos {\left (k - 4 \right )} \theta \left (k - 4\right )}{2 k} - \frac {\cos {\left (k + 4 \right )} \theta \left (- k - 4\right )}{2 k} + \frac {\theta \left (- k - 4\right )}{2 k} - \frac {\theta \left (k - 4\right )}{2 k}\right ) \cos {\left (t \right )} + \left (- \frac {\sin {\left (k - 4 \right )} \theta \left (k - 4\right )}{2 k} - \frac {\sin {\left (k + 4 \right )} \theta \left (- k - 4\right )}{2 k} + \frac {\cos {\left (k - 4 \right )} \delta \left (k - 4\right )}{2 k} - \frac {\cos {\left (k + 4 \right )} \delta \left (k + 4\right )}{2 k} - \frac {\delta \left (k - 4\right )}{2 k} + \frac {\delta \left (k + 4\right )}{2 k}\right ) \sin {\left (t \right )} + \frac {\cos {\left (k - t + 4 \right )} \theta \left (- k + t - 4\right )}{2 k} - \frac {\cos {\left (k + t - 4 \right )} \theta \left (k + t - 4\right )}{2 k} - \frac {\theta \left (- k + t - 4\right )}{2 k} + \frac {\theta \left (k + t - 4\right )}{2 k} \]