Internal problem ID [11930]
Internal file name [OUTPUT/11940_Saturday_April_13_2024_10_26_38_PM_75029156/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: 4.
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 )&=x \left (t \right )+2 y \left (t \right )+2 \,{\mathrm e}^{t}\\ x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=3 x \left (t \right )+4 y \left (t \right )+{\mathrm e}^{2 t} \end {align*}
The system is \begin {align*} x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=x \left (t \right )+2 y \left (t \right )+2 \,{\mathrm e}^{t}\tag {1}\\ x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=3 x \left (t \right )+4 y \left (t \right )+{\mathrm e}^{2 t}\tag {2} \end {align*}
Since the left side is the same, this implies \begin {align*} x \left (t \right )+2 y \left (t \right )+2 \,{\mathrm e}^{t}&=3 x \left (t \right )+4 y \left (t \right )+{\mathrm e}^{2 t}\\ y \left (t \right )&=-\frac {{\mathrm e}^{2 t}}{2}-x \left (t \right )+{\mathrm e}^{t}\tag {3} \end {align*}
Taking derivative of the above w.r.t. \(t\) gives \begin {align*} y^{\prime }\left (t \right )&=-{\mathrm e}^{2 t}-x^{\prime }\left (t \right )+{\mathrm e}^{t}\tag {4} \end {align*}
Substituting (3,4) in (1) to eliminate \(y \left (t \right ),y^{\prime }\left (t \right )\) gives \begin {align*} -{\mathrm e}^{2 t}+{\mathrm e}^{t} &= -x \left (t \right )-{\mathrm e}^{2 t}+4 \,{\mathrm e}^{t}\\ x \left (t \right ) &= 3 \,{\mathrm e}^{t}\tag {5} \end {align*}
Substituting (5) into (3) gives \begin {align*} y \left (t \right )&=-\frac {{\mathrm e}^{2 t}}{2}-2 \,{\mathrm e}^{t} \tag {7} \end {align*}
Since \(x \left (t \right ) = 3 \,{\mathrm e}^{t}\), is missing derivative in \(x\) then it is an algebraic equation. Solving for \(x \left (t \right )\). \begin {align*} \end {align*}
Failed to find solution. Terminating
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 23
dsolve([diff(x(t),t)+diff(y(t),t)-x(t)-2*y(t)=2*exp(t),diff(x(t),t)+diff(y(t),t)-3*x(t)-4*y(t)=exp(2*t)],singsol=all)
\begin{align*} x \left (t \right ) &= 3 \,{\mathrm e}^{t} \\ y \left (t \right ) &= -\frac {{\mathrm e}^{2 t}}{2}-2 \,{\mathrm e}^{t} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.013 (sec). Leaf size: 25
DSolve[{x'[t]+y'[t]-x[t]-2*y[t]==2*Exp[t],x'[t]+y'[t]-3*x[t]-4*y[t]==Exp[2*t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to 3 e^t \\ y(t)\to -\frac {1}{2} e^t \left (e^t+4\right ) \\ \end{align*}