63.15.14 problem 15

Internal problem ID [13126]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 3, Laplace transform. Section 3.2.1 Initial value problems. Exercises page 156
Problem number : 15
Date solved : Monday, March 31, 2025 at 07:34:56 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} x \left (0\right )&=1\\ x^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.352 (sec). Leaf size: 19

ode:=diff(diff(x(t),t),t)+Pi^2*x(t) = Pi^2*Heaviside(1-t); 
ic:=x(0) = 1, D(x)(0) = 0; 
dsolve([ode,ic],x(t),method='laplace');
 

\[ x = 1+\left (-\cos \left (\pi t \right )-1\right ) \operatorname {Heaviside}\left (t -1\right ) \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 18

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

\[ x(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 1 & t\leq 1 \\ -\cos (\pi t) & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 0.691 (sec). Leaf size: 22

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

\[ x{\left (t \right )} = \left (\theta \left (1 - t\right ) - 1\right ) \cos {\left (\pi t \right )} + \theta \left (1 - t\right ) \]