76.20.10 problem 10

Internal problem ID [17758]
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 : 10
Date solved : Tuesday, January 28, 2025 at 11:02:19 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 & \pi \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: 17.885 (sec). Leaf size: 58

dsolve([diff(y(t),t$2)+diff(y(t),t)+125/100*y(t)=piecewise(0<=t and t<Pi, sin(t), t>=Pi ,0),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.116 (sec). Leaf size: 77

DSolve[{D[y[t],{t,2}]+D[y[t],t]+125/100*y[t]==Piecewise[{  {Sin[t],0<= t <Pi}, {0,t>=Pi}}],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

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