Internal problem ID [11560]
Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag,
NY. 2015.
Section: Chapter 4, Linear Systems. Exercises page 218
Problem number: 2(d).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }&=x+y \left (t \right )\\ y^{\prime }\left (t \right )&=-3 x+3 y \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 76
dsolve([diff(x(t),t)=x(t)+y(t),diff(y(t),t)=-3*x(t)+3*y(t)],[x(t), y(t)], singsol=all)
\[ x \left (t \right ) = \frac {{\mathrm e}^{2 t} \left (\sin \left (\sqrt {2}\, t \right ) \sqrt {2}\, c_{2} -\cos \left (\sqrt {2}\, t \right ) \sqrt {2}\, c_{1} +c_{1} \sin \left (\sqrt {2}\, t \right )+c_{2} \cos \left (\sqrt {2}\, t \right )\right )}{3} \] \[ y \left (t \right ) = {\mathrm e}^{2 t} \left (c_{1} \sin \left (\sqrt {2}\, t \right )+c_{2} \cos \left (\sqrt {2}\, t \right )\right ) \]
✓ Solution by Mathematica
Time used: 0.027 (sec). Leaf size: 94
DSolve[{x'[t]==x[t]+y[t],y'[t]==-3*x[t]+3*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{2} e^{2 t} \left (2 c_1 \cos \left (\sqrt {2} t\right )+\sqrt {2} (c_2-c_1) \sin \left (\sqrt {2} t\right )\right ) y(t)\to \frac {1}{2} e^{2 t} \left (2 c_2 \cos \left (\sqrt {2} t\right )+\sqrt {2} (c_2-3 c_1) \sin \left (\sqrt {2} t\right )\right ) \end{align*}