5.4.9 problem 11(b)

Internal problem ID [1503]
Book : Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section : Chapter 6.4, The Laplace Transform. Differential equations with discontinuous forcing functions. page 268
Problem number : 11(b)
Date solved : Tuesday, September 30, 2025 at 04:34:49 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} u^{\prime \prime }+\frac {u^{\prime }}{4}+u&=k \left (\operatorname {Heaviside}\left (t -\frac {3}{2}\right )-\operatorname {Heaviside}\left (t -\frac {5}{2}\right )\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.186 (sec). Leaf size: 72

ode:=diff(diff(u(t),t),t)+1/4*diff(u(t),t)+u(t) = k*(Heaviside(t-3/2)-Heaviside(t-5/2)); 
ic:=[u(0) = 0, D(u)(0) = 0]; 
dsolve([ode,op(ic)],u(t),method='laplace');
 

\[ u = 2 k \left (\mathcal {L}^{-1}\left (\frac {\sinh \left (\frac {\textit {\_s1}}{2}\right ) {\mathrm e}^{-2 \textit {\_s1}}}{\textit {\_s1}}, \textit {\_s1} , t\right )-\mathcal {L}^{-1}\left (\frac {\sinh \left (\frac {\textit {\_s1}}{2}\right ) {\mathrm e}^{-2 \textit {\_s1}}}{4 \textit {\_s1}^{2}+\textit {\_s1} +4}, \textit {\_s1} , t\right )-4 \mathcal {L}^{-1}\left (\frac {\sinh \left (\frac {\textit {\_s1}}{2}\right ) {\mathrm e}^{-2 \textit {\_s1}} \textit {\_s1}}{4 \textit {\_s1}^{2}+\textit {\_s1} +4}, \textit {\_s1} , t\right )\right ) \]

✓ Mathematica. Time used: 0.073 (sec). Leaf size: 192

ode=D[u[t],{t,2}]+1/4*D[u[t],t]+u[t]==k*(UnitStep[t-3/2]-UnitStep[t-5/2]); 
ic={u[0]==0,Derivative[1][u][0]==0}; 
DSolve[{ode,ic},u[t],t,IncludeSingularSolutions->True]
 

\begin{align*} u(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} -e^{\frac {3}{16}-\frac {t}{8}} \cos \left (\frac {3}{16} \sqrt {7} (3-2 t)\right ) k+\frac {e^{\frac {3}{16}-\frac {t}{8}} \sin \left (\frac {3}{16} \sqrt {7} (3-2 t)\right ) k}{3 \sqrt {7}}+k & \frac {3}{2}<t\leq \frac {5}{2} \\ \frac {1}{21} e^{\frac {3}{16}-\frac {t}{8}} k \left (-21 \cos \left (\frac {3}{16} \sqrt {7} (3-2 t)\right )+21 \sqrt [8]{e} \cos \left (\frac {3}{16} \sqrt {7} (5-2 t)\right )+\sqrt {7} \left (\sin \left (\frac {3}{16} \sqrt {7} (3-2 t)\right )-\sqrt [8]{e} \sin \left (\frac {3}{16} \sqrt {7} (5-2 t)\right )\right )\right ) & 2 t>5 \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 4.032 (sec). Leaf size: 163

from sympy import * 
t = symbols("t") 
k = symbols("k") 
u = Function("u") 
ode = Eq(-k*(-Heaviside(t - 5/2) + Heaviside(t - 3/2)) + u(t) + Derivative(u(t), t)/4 + Derivative(u(t), (t, 2)),0) 
ics = {u(0): 0, Subs(Derivative(u(t), t), t, 0): 0} 
dsolve(ode,func=u(t),ics=ics)
 

\[ u{\left (t \right )} = - k \theta \left (t - \frac {5}{2}\right ) + k \theta \left (t - \frac {3}{2}\right ) + \left (\frac {\sqrt {7} k e^{\frac {5}{16}} \sin {\left (\frac {3 \sqrt {7} \left (2 t - 5\right )}{16} \right )} \theta \left (t - \frac {5}{2}\right )}{21} - \frac {\sqrt {7} k e^{\frac {3}{16}} \sin {\left (\frac {3 \sqrt {7} \left (2 t - 3\right )}{16} \right )} \theta \left (t - \frac {3}{2}\right )}{21} + k e^{\frac {5}{16}} \cos {\left (\frac {3 \sqrt {7} \left (2 t - 5\right )}{16} \right )} \theta \left (t - \frac {5}{2}\right ) - k e^{\frac {3}{16}} \cos {\left (\frac {3 \sqrt {7} \left (2 t - 3\right )}{16} \right )} \theta \left (t - \frac {3}{2}\right )\right ) e^{- \frac {t}{8}} \]