72.15.5 problem 19 (ii)

Internal problem ID [14886]
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 (ii)
Date solved : Tuesday, January 28, 2025 at 07:18:08 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-3 x \left (t \right )+y\\ y^{\prime }&=-x \left (t \right )+y \end{align*}

✓ Solution by Maple

Time used: 0.050 (sec). Leaf size: 81

dsolve([diff(x(t),t)=-3*x(t)+1*y(t),diff(y(t),t)=-1*x(t)+1*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\left (\sqrt {3}-1\right ) t}+c_{2} {\mathrm e}^{-\left (1+\sqrt {3}\right ) t} \\ y &= c_{1} {\mathrm e}^{\left (\sqrt {3}-1\right ) t} \sqrt {3}-c_{2} {\mathrm e}^{-\left (1+\sqrt {3}\right ) t} \sqrt {3}+2 c_{1} {\mathrm e}^{\left (\sqrt {3}-1\right ) t}+2 c_{2} {\mathrm e}^{-\left (1+\sqrt {3}\right ) t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 147

DSolve[{D[x[t],t]==-3*x[t]+1*y[t],D[y[t],t]==-1*x[t]+1*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{6} e^{-\left (\left (1+\sqrt {3}\right ) t\right )} \left (c_1 \left (\left (3-2 \sqrt {3}\right ) e^{2 \sqrt {3} t}+3+2 \sqrt {3}\right )+\sqrt {3} c_2 \left (e^{2 \sqrt {3} t}-1\right )\right ) \\ y(t)\to \frac {1}{6} e^{-\left (\left (1+\sqrt {3}\right ) t\right )} \left (c_2 \left (\left (3+2 \sqrt {3}\right ) e^{2 \sqrt {3} t}+3-2 \sqrt {3}\right )-\sqrt {3} c_1 \left (e^{2 \sqrt {3} t}-1\right )\right ) \\ \end{align*}