11.4.2 problem 2

Internal problem ID [1496]
Book : Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section : Chapter 6.4, The Laplace Transform. Differential equations with discontinuous forcing functions. page 268
Problem number : 2
Date solved : Monday, January 27, 2025 at 04:58:01 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 0.872 (sec). Leaf size: 64

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

\[ y = {\mathrm e}^{-t} \sin \left (t \right )+\frac {\left (\left \{\begin {array}{cc} 0 & t <\pi \\ 1+{\mathrm e}^{-t +\pi } \left (\cos \left (t \right )+\sin \left (t \right )\right ) & t <2 \pi \\ \left (\cos \left (t \right )+\sin \left (t \right )\right ) \left ({\mathrm e}^{2 \pi -t}+{\mathrm e}^{-t +\pi }\right ) & 2 \pi \le t \end {array}\right .\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.048 (sec). Leaf size: 89

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

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