Internal problem ID [13068]
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 3.1. page 258
Problem number: 8.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=-3 x \left (t \right )+2 \pi y\\ y^{\prime }&=4 x \left (t \right )-y \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 119
dsolve([diff(x(t),t)=-3*x(t)+2*Pi*y(t),diff(y(t),t)=4*x(t)-y(t)],singsol=all)
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{-\left (2+\sqrt {1+8 \pi }\right ) t}+c_{2} {\mathrm e}^{\left (-2+\sqrt {1+8 \pi }\right ) t} \\ y \left (t \right ) &= -\frac {c_{1} {\mathrm e}^{-\left (2+\sqrt {1+8 \pi }\right ) t} \sqrt {1+8 \pi }-c_{2} {\mathrm e}^{\left (-2+\sqrt {1+8 \pi }\right ) t} \sqrt {1+8 \pi }-c_{1} {\mathrm e}^{-\left (2+\sqrt {1+8 \pi }\right ) t}-c_{2} {\mathrm e}^{\left (-2+\sqrt {1+8 \pi }\right ) t}}{2 \pi } \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 189
DSolve[{x'[t]==-3*x[t]+2*Pi*y[t],y'[t]==4*x[t]-y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {e^{-\left (\left (2+\sqrt {1+8 \pi }\right ) t\right )} \left (c_1 \left (\left (\sqrt {1+8 \pi }-1\right ) e^{2 \sqrt {1+8 \pi } t}+1+\sqrt {1+8 \pi }\right )+2 \pi c_2 \left (e^{2 \sqrt {1+8 \pi } t}-1\right )\right )}{2 \sqrt {1+8 \pi }} \\ y(t)\to \frac {e^{-\left (\left (2+\sqrt {1+8 \pi }\right ) t\right )} \left (4 c_1 \left (e^{2 \sqrt {1+8 \pi } t}-1\right )+c_2 \left (\left (1+\sqrt {1+8 \pi }\right ) e^{2 \sqrt {1+8 \pi } t}-1+\sqrt {1+8 \pi }\right )\right )}{2 \sqrt {1+8 \pi }} \\ \end{align*}