20.22.9 problem Problem 35

Internal problem ID [3964]
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 35
Date solved : Tuesday, March 04, 2025 at 05:21:33 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y^{\prime }-2 y&=1-3 \operatorname {Heaviside}\left (t -2\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 3.247 (sec). Leaf size: 48

ode:=diff(diff(y(t),t),t)-diff(y(t),t)-2*y(t) = 1-3*Heaviside(t-2); 
ic:=y(0) = 1, D(y)(0) = -2; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = -\frac {1}{2}+\frac {5 \,{\mathrm e}^{-t}}{3}-\frac {{\mathrm e}^{2 t}}{6}-\frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{2 t -4}}{2}-\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-t +2}+\frac {3 \operatorname {Heaviside}\left (t -2\right )}{2} \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 70

ode=D[y[t],{t,2}]-D[y[t],t]-2*y[t]==1-3*UnitStep[t-2]; 
ic={y[0]==1,Derivative[1][y][0] ==-2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} -\frac {1}{6} e^{-t} \left (-10+3 e^t+e^{3 t}\right ) & t\leq 2 \\ \frac {1}{6} \left (6-6 e^{2-t}+10 e^{-t}-e^{2 t}-3 e^{2 t-4}\right ) & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 0.981 (sec). Leaf size: 54

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-2*y(t) + 3*Heaviside(t - 2) - Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 1,0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): -2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- \frac {\theta \left (t - 2\right )}{2 e^{4}} - \frac {1}{6}\right ) e^{2 t} + \left (- e^{2} \theta \left (t - 2\right ) + \frac {5}{3}\right ) e^{- t} + \frac {3 \theta \left (t - 2\right )}{2} - \frac {1}{2} \]