76.10.7 problem 7

Internal problem ID [17529]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.6 (A brief introduction to nonlinear systems). Problems at page 195
Problem number : 7
Date solved : Tuesday, January 28, 2025 at 10:39:59 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )-y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-2 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.059 (sec). Leaf size: 69

dsolve([diff(x(t),t)=2*x(t)-y(t),diff(y(t),t)=x(t)-2*y(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 144

DSolve[{D[x[t],t]==2*x[t]-y[t],D[y[t],t]==x[t]-2*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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