76.7.1 problem 1

Internal problem ID [17474]
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 3.3 (Homogeneous linear systems with constant coefficients). Problems at page 165
Problem number : 1
Date solved : Tuesday, January 28, 2025 at 10:39:15 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=3 x \left (t \right )-2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )-2 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.066 (sec). Leaf size: 35

dsolve([diff(x(t),t)=3*x(t)-2*y(t),diff(y(t),t)=2*x(t)-2*y(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 73

DSolve[{D[x[t],t]==3*x[t]-2*y[t],D[y[t],t]==2*x[t]-2*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{3} e^{-t} \left (c_1 \left (4 e^{3 t}-1\right )-2 c_2 \left (e^{3 t}-1\right )\right ) \\ y(t)\to \frac {1}{3} e^{-t} \left (2 c_1 \left (e^{3 t}-1\right )-c_2 \left (e^{3 t}-4\right )\right ) \\ \end{align*}