64.24.1 problem 1

Internal problem ID [13686]
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 : 1
Date solved : Tuesday, January 28, 2025 at 05:55:15 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )+3 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=3 x \left (t \right )+y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.057 (sec). Leaf size: 34

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

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{-2 t}+c_{2} {\mathrm e}^{4 t} \\ y \left (t \right ) &= -c_{1} {\mathrm e}^{-2 t}+c_{2} {\mathrm e}^{4 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 68

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

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