Internal problem ID [13084]
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 3.2. page 277
Problem number: 9.
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 )+y \end {align*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 86
dsolve([diff(x(t),t)=2*x(t)+y(t),diff(y(t),t)=x(t)+y(t)],singsol=all)
\begin{align*} x \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}} \\ y \left (t \right ) &= \frac {c_{1} {\mathrm e}^{\frac {\left (3+\sqrt {5}\right ) t}{2}} \sqrt {5}}{2}-\frac {c_{2} {\mathrm e}^{-\frac {\left (\sqrt {5}-3\right ) t}{2}} \sqrt {5}}{2}-\frac {c_{1} {\mathrm e}^{\frac {\left (3+\sqrt {5}\right ) t}{2}}}{2}-\frac {c_{2} {\mathrm e}^{-\frac {\left (\sqrt {5}-3\right ) t}{2}}}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 145
DSolve[{x'[t]==2*x[t]+y[t],y'[t]==x[t]+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+\sqrt {5}\right ) e^{\sqrt {5} t}+5-\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 (\sqrt {5}-5\right ) e^{\sqrt {5} t}-5-\sqrt {5}\right )\right ) \\ \end{align*}