14.22.10 problem Problem 36

Internal problem ID [3965]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 10, The Laplace Transform and Some Elementary Applications. Exercises for 10.7. page 704
Problem number : Problem 36
Date solved : Tuesday, September 30, 2025 at 06:59:40 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y&=\operatorname {Heaviside}\left (t -1\right )-\operatorname {Heaviside}\left (t -2\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=4 \\ \end{align*}

✓ Maple. Time used: 0.300 (sec). Leaf size: 35

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

\[ y = \frac {\operatorname {Heaviside}\left (t -1\right ) \sinh \left (t -1\right )^{2}}{2}-\frac {\operatorname {Heaviside}\left (t -2\right ) \sinh \left (t -2\right )^{2}}{2}+2 \sinh \left (2 t \right ) \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 113

ode=D[y[t],{t,2}]-4*y[t]==UnitStep[t-1]-UnitStep[t-2]; 
ic={y[0]==0,Derivative[1][y][0] ==4}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-2 t} \left (-1+e^{4 t}\right ) & t\leq 1 \\ \frac {1}{8} \left (-2+e^{2-2 t}-8 e^{-2 t}+8 e^{2 t}+e^{2 t-2}\right ) & 1<t\leq 2 \\ \frac {1}{8} e^{-2 (t+2)} \left (-8 e^4+e^6-e^8-e^{4 t}+e^{4 t+2}+8 e^{4 t+4}\right ) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.936 (sec). Leaf size: 80

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-4*y(t) + Heaviside(t - 2) - Heaviside(t - 1) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 4} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- \frac {\theta \left (t - 2\right )}{8 e^{4}} + \frac {\theta \left (t - 1\right )}{8 e^{2}} + 1\right ) e^{2 t} + \left (- \frac {e^{4} \theta \left (t - 2\right )}{8} + \frac {e^{2} \theta \left (t - 1\right )}{8} - 1\right ) e^{- 2 t} + \frac {\theta \left (t - 2\right )}{4} - \frac {\theta \left (t - 1\right )}{4} \]