76.20.6 problem 6

Internal problem ID [17754]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.6 (Differential equations with Discontinuous Forcing Functions). Problems at page 342
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 11:02:15 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=6\\ y^{\prime }\left (0\right )&=6 \end{align*}

✓ Solution by Maple

Time used: 12.308 (sec). Leaf size: 47

dsolve([diff(y(t),t$2)+3*diff(y(t),t)+2*y(t)=Heaviside(t-2),y(0) = 6, D(y)(0) = 6],y(t), singsol=all)
 

\[ y = -12 \,{\mathrm e}^{-2 t}+18 \,{\mathrm e}^{-t}-\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-t +2}+\frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-2 t +4}}{2}+\frac {\operatorname {Heaviside}\left (t -2\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.027 (sec). Leaf size: 55

DSolve[{D[y[t],{t,2}]+3*D[y[t],t]+2*y[t]==UnitStep[t-2],{y[0]==6,Derivative[1][y][0] ==6}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{2} e^{-2 t} \left (\left (e^2-e^t\right )^2 (-\theta (2-t))+36 e^t+e^{2 t}-2 e^{t+2}+e^4-24\right ) \]