Internal problem ID [13113]
Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th
edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 3. Linear Systems. Exercises section 3.5 page 327
Problem number: 2.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+y\\ y^{\prime }&=-x \left (t \right )-2 y \end {align*}
With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 0] \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 106
dsolve([diff(x(t),t) = 2*x(t)+y(t), diff(y(t),t) = -x(t)-2*y(t), x(0) = 1, y(0) = 0], singsol=all)
\begin{align*} x \left (t \right ) &= \left (\frac {1}{2}+\frac {\sqrt {3}}{3}\right ) {\mathrm e}^{\sqrt {3}\, t}+\left (\frac {1}{2}-\frac {\sqrt {3}}{3}\right ) {\mathrm e}^{-\sqrt {3}\, t} \\ y \left (t \right ) &= \left (\frac {1}{2}+\frac {\sqrt {3}}{3}\right ) \sqrt {3}\, {\mathrm e}^{\sqrt {3}\, t}-\left (\frac {1}{2}-\frac {\sqrt {3}}{3}\right ) \sqrt {3}\, {\mathrm e}^{-\sqrt {3}\, t}-2 \left (\frac {1}{2}+\frac {\sqrt {3}}{3}\right ) {\mathrm e}^{\sqrt {3}\, t}-2 \left (\frac {1}{2}-\frac {\sqrt {3}}{3}\right ) {\mathrm e}^{-\sqrt {3}\, t} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.018 (sec). Leaf size: 82
DSolve[{x'[t]==2*x[t]+1*y[t],y'[t]==-1*x[t]-2*y[t]},{x[0]==1,y[0]==0},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{6} e^{-\sqrt {3} t} \left (\left (3+2 \sqrt {3}\right ) e^{2 \sqrt {3} t}+3-2 \sqrt {3}\right ) \\ y(t)\to -\frac {e^{-\sqrt {3} t} \left (e^{2 \sqrt {3} t}-1\right )}{2 \sqrt {3}} \\ \end{align*}