64.16.9 problem 9

Internal problem ID [13612]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 7, Systems of linear differential equations. Section 7.1. Exercises page 277
Problem number : 9
Date solved : Tuesday, January 28, 2025 at 05:53:52 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+\frac {d}{d t}y \left (t \right )+2 y \left (t \right )&=\sin \left (t \right )\\ \frac {d}{d t}x \left (t \right )+\frac {d}{d t}y \left (t \right )-x \left (t \right )-y \left (t \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.257 (sec). Leaf size: 26

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

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{t}-\frac {\sin \left (t \right )}{2} \\ y \left (t \right ) &= -\frac {c_{1} {\mathrm e}^{t}}{3}+\frac {\sin \left (t \right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 38

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

\begin{align*} x(t)\to \frac {1}{2} \left (-\sin (t)+3 c_1 e^t\right ) \\ y(t)\to \frac {1}{2} \left (\sin (t)-c_1 e^t\right ) \\ \end{align*}