11.4.6 problem 6

Internal problem ID [1500]
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 : 6
Date solved : Monday, January 27, 2025 at 04:58:06 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+\frac {5 y}{4}&=\left \{\begin {array}{cc} \sin \left (t \right ) & 0\le t <\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 )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.555 (sec). Leaf size: 58

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

\[ y = \frac {4 \left (\left \{\begin {array}{cc} -8 \,{\mathrm e}^{-\frac {t}{4}} \left (\cos \left (t \right ) \sinh \left (\frac {t}{4}\right )-\frac {\sin \left (t \right ) \cosh \left (\frac {t}{4}\right )}{4}\right ) & t <\pi \\ \left (\sin \left (t \right )+4 \cos \left (t \right )\right ) \left ({\mathrm e}^{-\frac {t}{2}}-{\mathrm e}^{-\frac {t}{2}+\frac {\pi }{2}}\right ) & \pi \le t \end {array}\right .\right )}{17} \]

✓ Solution by Mathematica

Time used: 0.120 (sec). Leaf size: 77

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

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 0 & t\leq 0 \\ \frac {4}{17} \left (\left (-4+4 e^{-t/2}\right ) \cos (t)+\left (1+e^{-t/2}\right ) \sin (t)\right ) & 0<t\leq \pi \\ -\frac {4}{17} e^{-t/2} \left (-1+e^{\pi /2}\right ) (4 \cos (t)+\sin (t)) & \text {True} \\ \end {array} \\ \end {array} \]