20.22.11 problem Problem 37

Internal problem ID [3966]
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 37
Date solved : Sunday, March 30, 2025 at 02:13:07 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.337 (sec). Leaf size: 24

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

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

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 31

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

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} t+2 \cos (t) & t\leq 1 \\ 2 \cos (t)-\sin (1-t)+1 & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 1.089 (sec). Leaf size: 36

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

\[ y{\left (t \right )} = - t \theta \left (t - 1\right ) + t + \sin {\left (t - 1 \right )} \theta \left (t - 1\right ) + 2 \cos {\left (t \right )} + \theta \left (t - 1\right ) \]