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 : Monday, January 27, 2025 at 06:06:23 AM
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*}

✓ Solution by Maple

Time used: 1.185 (sec). Leaf size: 39

dsolve([diff(y(t),t$2)+2*diff(y(t),t)+y(t)=2*(t-3)*Heaviside(t-3),y(0) = 2, D(y)(0) = 1],y(t), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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