76.22.4 problem 17

Internal problem ID [17785]
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 5.8 (Convolution Integrals and Their Applications). Problems at page 359
Problem number : 17
Date solved : Tuesday, January 28, 2025 at 11:03:02 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+\frac {5 y}{4}&=1-\operatorname {Heaviside}\left (t -\pi \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 13.036 (sec). Leaf size: 52

dsolve([diff(y(t),t$2)+diff(y(t),t)+125/100*y(t)=1-Heaviside(t-Pi),y(0) = 1, D(y)(0) = -1],y(t), singsol=all)
 

\[ y = -\frac {4 \left (\cos \left (t \right )+\frac {\sin \left (t \right )}{2}\right ) \operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{-\frac {t}{2}+\frac {\pi }{2}}}{5}-\frac {4 \operatorname {Heaviside}\left (t -\pi \right )}{5}+\frac {4}{5}+\frac {{\mathrm e}^{-\frac {t}{2}} \left (2 \cos \left (t \right )-9 \sin \left (t \right )\right )}{10} \]

✓ Solution by Mathematica

Time used: 0.033 (sec). Leaf size: 80

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

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{10} \left (2 e^{-t/2} \cos (t)-9 e^{-t/2} \sin (t)+8\right ) & t\leq \pi \\ \frac {1}{10} e^{-t/2} \left (\left (2-8 e^{\pi /2}\right ) \cos (t)-\left (9+4 e^{\pi /2}\right ) \sin (t)\right ) & \text {True} \\ \end {array} \\ \end {array} \]