64.24.7 problem 7

Internal problem ID [13692]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 13, Nonlinear differential equations. Section 13.2, Exercises page 656
Problem number : 7
Date solved : Tuesday, January 28, 2025 at 05:55:20 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )-y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+5 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.039 (sec). Leaf size: 81

dsolve([diff(x(t),t)=x(t)-y(t),diff(y(t),t)=x(t)+5*y(t)],singsol=all)
 

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

✓ Solution by Mathematica

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

DSolve[{D[x[t],t]==x[t]-y[t],D[y[t],t]==x[t]+5*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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