14.18.1 problem 1

Internal problem ID [2685]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.11, Differential equations with discontinuous right-hand sides. Excercises page 243
Problem number : 1
Date solved : Tuesday, March 04, 2025 at 02:34:05 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+y&=2 \left (t -3\right ) \operatorname {Heaviside}\left (t -3\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: 1.185 (sec). Leaf size: 39

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

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

✓ Mathematica. Time used: 0.036 (sec). Leaf size: 49

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

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-t} (3 t+2) & t\leq 3 \\ e^{-t} \left (2 e^t (t-5)+2 e^3 (t-1)+3 t+2\right ) & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 1.496 (sec). Leaf size: 53

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq((6 - 2*t)*Heaviside(t - 3) + y(t) + 2*Derivative(y(t), 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)
 

\[ y{\left (t \right )} = 2 t \theta \left (t - 3\right ) + \left (t \left (2 e^{3} \theta \left (t - 3\right ) + 3\right ) - 2 e^{3} \theta \left (t - 3\right ) + 2\right ) e^{- t} - 10 \theta \left (t - 3\right ) \]