Internal problem ID [11935]
Internal file name [OUTPUT/11945_Sunday_April_14_2024_02_31_07_AM_5011505/index.tex]
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.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "system of linear ODEs"
Solve \begin {align*} x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=-2 y \left (t \right )+\sin \left (t \right )\\ x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right ) \end {align*}
The system is \begin {align*} x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=-2 y \left (t \right )+\sin \left (t \right )\tag {1}\\ x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right )\tag {2} \end {align*}
Since the left side is the same, this implies \begin {align*} -2 y \left (t \right )+\sin \left (t \right )&=x \left (t \right )+y \left (t \right )\\ y \left (t \right )&=-\frac {x \left (t \right )}{3}+\frac {\sin \left (t \right )}{3}\tag {3} \end {align*}
Taking derivative of the above w.r.t. \(t\) gives \begin {align*} y^{\prime }\left (t \right )&=-\frac {x^{\prime }\left (t \right )}{3}+\frac {\cos \left (t \right )}{3}\tag {4} \end {align*}
Substituting (3,4) in (1) to eliminate \(y \left (t \right ),y^{\prime }\left (t \right )\) gives \begin {align*} \frac {2 x^{\prime }\left (t \right )}{3}+\frac {\cos \left (t \right )}{3} &= \frac {2 x \left (t \right )}{3}+\frac {\sin \left (t \right )}{3}\\ x^{\prime }\left (t \right ) &= x \left (t \right )+\frac {\sin \left (t \right )}{2}-\frac {\cos \left (t \right )}{2}\tag {5} \end {align*}
Which is now solved for \(x \left (t \right )\).
Entering Linear first order ODE solver. In canonical form a linear first order is \begin {align*} x^{\prime }\left (t \right ) + p(t)x \left (t \right ) &= q(t) \end {align*}
Where here \begin {align*} p(t) &=-1\\ q(t) &=\frac {\sin \left (t \right )}{2}-\frac {\cos \left (t \right )}{2} \end {align*}
Hence the ode is \begin {align*} -x \left (t \right )+x^{\prime }\left (t \right ) = \frac {\sin \left (t \right )}{2}-\frac {\cos \left (t \right )}{2} \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int \left (-1\right )d t} \\ &= {\mathrm e}^{-t} \\ \end{align*} The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}t}}\left ( \mu x\right ) &= \left (\mu \right ) \left (\frac {\sin \left (t \right )}{2}-\frac {\cos \left (t \right )}{2}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}t}} \left ({\mathrm e}^{-t} x\right ) &= \left ({\mathrm e}^{-t}\right ) \left (\frac {\sin \left (t \right )}{2}-\frac {\cos \left (t \right )}{2}\right )\\ \mathrm {d} \left ({\mathrm e}^{-t} x\right ) &= \left (\frac {\left (\sin \left (t \right )-\cos \left (t \right )\right ) {\mathrm e}^{-t}}{2}\right )\, \mathrm {d} t \end {align*}
Integrating gives \begin {align*} {\mathrm e}^{-t} x &= \int {\frac {\left (\sin \left (t \right )-\cos \left (t \right )\right ) {\mathrm e}^{-t}}{2}\,\mathrm {d} t}\\ {\mathrm e}^{-t} x &= -\frac {{\mathrm e}^{-t} \sin \left (t \right )}{2} + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu ={\mathrm e}^{-t}\) results in \begin {align*} x \left (t \right ) &= -\frac {{\mathrm e}^{t} {\mathrm e}^{-t} \sin \left (t \right )}{2}+{\mathrm e}^{t} c_{1} \end {align*}
which simplifies to \begin {align*} x \left (t \right ) &= {\mathrm e}^{t} c_{1} -\frac {\sin \left (t \right )}{2} \end {align*}
Given now that we have the solution \begin {align*} x \left (t \right )&={\mathrm e}^{t} c_{1} -\frac {\sin \left (t \right )}{2} \tag {6} \end {align*}
Then substituting (6) into (3) gives \begin {align*} y \left (t \right )&=-\frac {{\mathrm e}^{t} c_{1}}{3}+\frac {\sin \left (t \right )}{2} \tag {7} \end {align*}
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 27
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: 0.02 (sec). Leaf size: 38
DSolve[{x'[t]+y'[t]+2*y[t]==Sin[t],x'[t]+y'[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*}