46.7.9 problem 26

Internal problem ID [7370]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 6. Laplace Transforms. Problem set 6.3, page 224
Problem number : 26
Date solved : Monday, January 27, 2025 at 02:51:08 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+5 y&=\left \{\begin {array}{cc} 10 \sin \left (t \right ) & 0<t <2 \pi \\ 0 & 2 \pi <t \end {array}\right . \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (\pi \right )&=1\\ y^{\prime }\left (\pi \right )&=2 \,{\mathrm e}^{-\pi }-2 \end{align*}

✓ Solution by Maple

Time used: 0.466 (sec). Leaf size: 70

dsolve([diff(y(t),t$2)+2*diff(y(t),t)+5*y(t)=piecewise(0<t and t<2*Pi,10*sin(t),t>2*Pi,0),y(Pi) = 1, D(y)(Pi) = 2*exp(-Pi)-2],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.062 (sec). Leaf size: 94

DSolve[{D[y[t],{t,2}]+2*D[y[t],t]+5*y[t]==Piecewise[{{10*Sin[t],0<t<2*Pi},{0,t>2*Pi}}],{y[Pi]==1,Derivative[1][y][Pi]==2*Exp[-Pi]-2}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{2} e^{-t} (3 \sin (2 t)-2 \cos (2 t)) & t\leq 0 \\ -\cos (t)+2 \sin (t)+e^{-t} \sin (2 t) & 0<t\leq 2 \pi \\ \frac {1}{2} e^{-t} \left (\left (2+e^{2 \pi }\right ) \sin (2 t)-2 e^{2 \pi } \cos (2 t)\right ) & \text {True} \\ \end {array} \\ \end {array} \]