76.20.2 problem 2

Internal problem ID [17750]
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 : 2
Date solved : Tuesday, January 28, 2025 at 11:02:10 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=5\\ y^{\prime }\left (0\right )&=4 \end{align*}

✓ Solution by Maple

Time used: 15.780 (sec). Leaf size: 103

dsolve([diff(y(t),t$2)+2*diff(y(t),t)+2*y(t)=piecewise(0<=t and t<Pi, 0, t>=Pi and t<=2*Pi ,1, t<=2*Pi,0),y(0) = 5, D(y)(0) = 4],y(t), singsol=all)
 

\[ y = \left \{\begin {array}{cc} {\mathrm e}^{-t} \left (5 \cos \left (t \right )+9 \sin \left (t \right )\right ) & t <\pi \\ {\mathrm e}^{-t} \left (5 \cos \left (t \right )+9 \sin \left (t \right )\right )+\frac {1}{2}+\frac {{\mathrm e}^{\pi -t} \left (\cos \left (t \right )+\sin \left (t \right )\right )}{2} & t <2 \pi \\ \frac {{\mathrm e}^{2 \pi -t} \left (\cos \left (t \right )+\sin \left (t \right )\right )}{2}+\frac {{\mathrm e}^{\pi -t} \left (\cos \left (t \right )+\sin \left (t \right )\right )}{2}+5 \,{\mathrm e}^{-t} \left (\cos \left (t \right )+\frac {9 \sin \left (t \right )}{5}\right ) & 2 \pi \le t \end {array}\right . \]

✓ Solution by Mathematica

Time used: 0.046 (sec). Leaf size: 100

DSolve[{D[y[t],{t,2}]+2*D[y[t],t]+2*y[t]==Piecewise[{  {0,0<= t <Pi}, {1,Pi<= t <=2*Pi}, {0, t>=2*Pi}}],{y[0]==5,Derivative[1][y][0] ==4}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-t} (5 \cos (t)+9 \sin (t)) & t\leq \pi \\ \frac {1}{2} e^{-t} \left (\left (10+e^{\pi }\right ) \cos (t)+e^t+\left (18+e^{\pi }\right ) \sin (t)\right ) & \pi <t\leq 2 \pi \\ \frac {1}{2} e^{-t} \left (\left (10+e^{\pi }+e^{2 \pi }\right ) \cos (t)+\left (18+e^{\pi }+e^{2 \pi }\right ) \sin (t)\right ) & \text {True} \\ \end {array} \\ \end {array} \]