28.4.28 problem 7.28

Internal problem ID [4560]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 7. Systems of linear differential equations. Problems at page 351
Problem number : 7.28
Date solved : Monday, January 27, 2025 at 09:23:29 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )-x \left (t \right )+y&=2 \sin \left (t \right ) \left (1-\operatorname {Heaviside}\left (t -\pi \right )\right )\\ 2 x \left (t \right )-y^{\prime }-y&=0 \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 0\\ y \left (0\right ) = 0 \end{align*}

✓ Solution by Maple

Time used: 0.053 (sec). Leaf size: 113

dsolve([diff(x(t),t)-x(t)+y(t) = 2*sin(t)*(1-Heaviside(t-Pi)), 2*x(t)-diff(y(t),t)-y(t) = 0, x(0) = 0, y(0) = 0], singsol=all)
 

\begin{align*} x &= -\cos \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right ) \pi +\sin \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right ) \pi +\cos \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right ) t -\sin \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right ) t -\cos \left (t \right ) t -\sin \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right )+\sin \left (t \right ) t +\sin \left (t \right ) \\ y &= -2 \cos \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right ) \pi +2 \cos \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right ) t -2 \cos \left (t \right ) t -2 \sin \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right )+2 \sin \left (t \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.116 (sec). Leaf size: 56

DSolve[{D[x[t],t]-x[t]+y[t]==2*Sin[t]*(1-UnitStep[t-Pi]),2*x[t]-D[y[t],t]-y[t]==0},{x[0]==0,y[0]==0},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \pi (\sin (t)-\cos (t)) & t\geq \pi \\ (t+1) \sin (t)-t \cos (t) & \text {True} \\ \end {array} \\ \end {array} \\ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} -2 \pi \cos (t) & t\geq \pi \\ 2 \sin (t)-2 t \cos (t) & \text {True} \\ \end {array} \\ \end {array} \\ \end{align*}