76.20.5 problem 5

Internal problem ID [17753]
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 : 5
Date solved : Tuesday, January 28, 2025 at 11:02:14 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=\left \{\begin {array}{cc} 1 & 0\le t <10 \\ 0 & 10\le t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 18.807 (sec). Leaf size: 63

dsolve([diff(y(t),t$2)+3*diff(y(t),t)+2*y(t)=piecewise(0<=t and t<10, 1, t>=10 ,0),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \frac {\left (\left \{\begin {array}{cc} 1-2 \,{\mathrm e}^{-t}+{\mathrm e}^{-2 t} & t <10 \\ {\mathrm e}^{-20}-2 \,{\mathrm e}^{-10}+2 & t =10 \\ 2 \,{\mathrm e}^{-t +10}-{\mathrm e}^{-2 t +20}-2 \,{\mathrm e}^{-t}+{\mathrm e}^{-2 t} & 10<t \end {array}\right .\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.039 (sec). Leaf size: 61

DSolve[{D[y[t],{t,2}]+3*D[y[t],t]+2*y[t]==Piecewise[{  {1,0<= t <10}, {0,t>=10}}],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 0 & t\leq 0 \\ \frac {1}{2} e^{-2 t} \left (-1+e^t\right )^2 & 0<t\leq 10 \\ \frac {1}{2} e^{-2 t} \left (-1+e^{10}\right ) \left (-1-e^{10}+2 e^t\right ) & \text {True} \\ \end {array} \\ \end {array} \]