Internal problem ID [12236]
Book: Nonlinear Ordinary Differential Equations by D.W.Jordna and P.Smith. 4th edition 1999.
Oxford Univ. Press. NY
Section: Chapter 2. Plane autonomous systems and linearization. Problems page 79
Problem number: 2.1 (ii).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )-2 y \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 86
dsolve([diff(x(t),t)=x(t)+y(t),diff(y(t),t)=x(t)-2*y(t)],[x(t), y(t)], singsol=all)
\[ x \left (t \right ) = \frac {c_{1} {\mathrm e}^{\frac {\left (-1+\sqrt {13}\right ) t}{2}} \sqrt {13}}{2}-\frac {c_{2} {\mathrm e}^{-\frac {\left (1+\sqrt {13}\right ) t}{2}} \sqrt {13}}{2}+\frac {3 c_{1} {\mathrm e}^{\frac {\left (-1+\sqrt {13}\right ) t}{2}}}{2}+\frac {3 c_{2} {\mathrm e}^{-\frac {\left (1+\sqrt {13}\right ) t}{2}}}{2} \] \[ y \left (t \right ) = c_{1} {\mathrm e}^{\frac {\left (-1+\sqrt {13}\right ) t}{2}}+c_{2} {\mathrm e}^{-\frac {\left (1+\sqrt {13}\right ) t}{2}} \]
✓ Solution by Mathematica
Time used: 0.017 (sec). Leaf size: 149
DSolve[{x'[t]==x[t]+y[t],y'[t]==x[t]-2*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{26} e^{-\frac {1}{2} \left (1+\sqrt {13}\right ) t} \left (c_1 \left (\left (13+3 \sqrt {13}\right ) e^{\sqrt {13} t}+13-3 \sqrt {13}\right )+2 \sqrt {13} c_2 \left (e^{\sqrt {13} t}-1\right )\right ) y(t)\to \frac {1}{26} e^{-\frac {1}{2} \left (1+\sqrt {13}\right ) t} \left (2 \sqrt {13} c_1 \left (e^{\sqrt {13} t}-1\right )-c_2 \left (\left (3 \sqrt {13}-13\right ) e^{\sqrt {13} t}-13-3 \sqrt {13}\right )\right ) \end{align*}