Internal problem ID [12830]
Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th
edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 3. Linear Systems. Review Exercises for chapter 3. page 376
Problem number: 19 (ii).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=-3 x \left (t \right )+y\\ y^{\prime }&=-x \left (t \right )+y \end {align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 82
dsolve([diff(x(t),t)=-3*x(t)+1*y(t),diff(y(t),t)=-1*x(t)+1*y(t)],[x(t), y(t)], singsol=all)
\begin{align*} x \left (t \right ) = -c_{1} {\mathrm e}^{\left (\sqrt {3}-1\right ) t} \sqrt {3}+c_{2} {\mathrm e}^{-\left (1+\sqrt {3}\right ) t} \sqrt {3}+2 c_{1} {\mathrm e}^{\left (\sqrt {3}-1\right ) t}+2 c_{2} {\mathrm e}^{-\left (1+\sqrt {3}\right ) t} y \left (t \right ) = c_{1} {\mathrm e}^{\left (\sqrt {3}-1\right ) t}+c_{2} {\mathrm e}^{-\left (1+\sqrt {3}\right ) t} \end{align*}
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 147
DSolve[{x'[t]==-3*x[t]+1*y[t],y'[t]==-1*x[t]+1*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{6} e^{-\left (\left (1+\sqrt {3}\right ) t\right )} \left (c_1 \left (\left (3-2 \sqrt {3}\right ) e^{2 \sqrt {3} t}+3+2 \sqrt {3}\right )+\sqrt {3} c_2 \left (e^{2 \sqrt {3} t}-1\right )\right ) y(t)\to \frac {1}{6} e^{-\left (\left (1+\sqrt {3}\right ) t\right )} \left (c_2 \left (\left (3+2 \sqrt {3}\right ) e^{2 \sqrt {3} t}+3-2 \sqrt {3}\right )-\sqrt {3} c_1 \left (e^{2 \sqrt {3} t}-1\right )\right ) \end{align*}