57.16.1 problem 7

Internal problem ID [14479]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 3, Laplace transform. Section 3.3 The convolution property. Exercises page 162
Problem number : 7
Date solved : Thursday, October 02, 2025 at 09:37:41 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.185 (sec). Leaf size: 24

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

\[ x = -\frac {1}{4}+\frac {\cosh \left (2 t \right )}{4}-\frac {\operatorname {Heaviside}\left (t -1\right ) \sinh \left (t -1\right )^{2}}{2} \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 54

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

\begin{align*} x(t)&\to \frac {1}{8} e^{-2 (t+1)} \left (\left (e^2-e^{2 t}\right )^2 \theta (1-t)+\left (e^2-1\right ) \left (e^{4 t}-e^2\right )\right ) \end{align*}

✓ Sympy. Time used: 0.443 (sec). Leaf size: 54

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

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