Internal problem ID [10521]
Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY.
2015.
Section: Chapter 4, Linear Systems. Exercises page 202
Problem number: 1(a).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=-2 x \left (t \right )-3 y \left (t \right )\\ y^{\prime }\left (t \right )&=-x \left (t \right )+4 y \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 95
dsolve([diff(x(t),t)=-2*x(t)-3*y(t),diff(y(t),t)=-x(t)+4*y(t)],[x(t), y(t)], singsol=all)
\[ x \left (t \right ) = -2 c_{1} {\mathrm e}^{\left (1+2 \sqrt {3}\right ) t} \sqrt {3}+2 c_{2} {\mathrm e}^{-\left (-1+2 \sqrt {3}\right ) t} \sqrt {3}+3 c_{1} {\mathrm e}^{\left (1+2 \sqrt {3}\right ) t}+3 c_{2} {\mathrm e}^{-\left (-1+2 \sqrt {3}\right ) t} \] \[ y \left (t \right ) = c_{1} {\mathrm e}^{\left (1+2 \sqrt {3}\right ) t}+c_{2} {\mathrm e}^{-\left (-1+2 \sqrt {3}\right ) t} \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 94
DSolve[{x'[t]==-2*x[t]-3*y[t],y'[t]==-x[t]+4*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{2} e^t \left (2 c_1 \cosh \left (2 \sqrt {3} t\right )-\sqrt {3} (c_1+c_2) \sinh \left (2 \sqrt {3} t\right )\right ) \\ y(t)\to \frac {1}{6} e^t \left (6 c_2 \cosh \left (2 \sqrt {3} t\right )-\sqrt {3} (c_1-3 c_2) \sinh \left (2 \sqrt {3} t\right )\right ) \\ \end{align*}