Internal problem ID [12834]
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 (vi).
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 ) \end {align*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 68
dsolve([diff(x(t),t)=3*x(t)+1*y(t),diff(y(t),t)=-1*x(t)+0*y(t)],[x(t), y(t)], singsol=all)
\begin{align*} x \left (t \right ) = \left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) c_{2} {\mathrm e}^{-\frac {\left (\sqrt {5}-3\right ) t}{2}}+\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) c_{1} {\mathrm e}^{\frac {\left (3+\sqrt {5}\right ) t}{2}} y \left (t \right ) = c_{1} {\mathrm e}^{\frac {\left (3+\sqrt {5}\right ) t}{2}}+c_{2} {\mathrm e}^{-\frac {\left (\sqrt {5}-3\right ) t}{2}} \end{align*}
✓ Solution by Mathematica
Time used: 0.018 (sec). Leaf size: 148
DSolve[{x'[t]==3*x[t]+1*y[t],y'[t]==-1*x[t]+0*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{10} e^{-\frac {1}{2} \left (\sqrt {5}-3\right ) t} \left (c_1 \left (\left (5+3 \sqrt {5}\right ) e^{\sqrt {5} t}+5-3 \sqrt {5}\right )+2 \sqrt {5} c_2 \left (e^{\sqrt {5} t}-1\right )\right ) y(t)\to -\frac {1}{10} e^{-\frac {1}{2} \left (\sqrt {5}-3\right ) t} \left (2 \sqrt {5} c_1 \left (e^{\sqrt {5} t}-1\right )+c_2 \left (\left (3 \sqrt {5}-5\right ) e^{\sqrt {5} t}-5-3 \sqrt {5}\right )\right ) \end{align*}