67.4.19 problem Problem 3(e)

Internal problem ID [14063]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 3(e)
Date solved : Tuesday, January 28, 2025 at 06:13:24 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+2 y&=5 \cos \left (t \right ) \left (\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )\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: 11.174 (sec). Leaf size: 58

dsolve([diff(y(t),t$2)+2*diff(y(t),t)+2*y(t)=5*cos(t)*(Heaviside(t)-Heaviside(t-Pi/2)),y(0) = 1, D(y)(0) = -1],y(t), singsol=all)
 

\[ y = -\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (\cos \left (t \right )-2 \sin \left (t \right )\right ) {\mathrm e}^{\frac {\pi }{2}-t}+\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (-\cos \left (t \right )-2 \sin \left (t \right )\right )-3 \,{\mathrm e}^{-t} \sin \left (t \right )+\cos \left (t \right )+2 \sin \left (t \right ) \]

✓ Solution by Mathematica

Time used: 0.052 (sec). Leaf size: 72

DSolve[{D[y[t],{t,2}]+2*D[y[t],t]+2*y[t]==5*Cos[t]*(UnitStep[t]-UnitStep[t-Pi/2]),{y[0]==1,Derivative[1][y][0] ==-1}},y[t],t,IncludeSingularSolutions -> True]
 

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