76.7.2 problem 2

Internal problem ID [17475]
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 : 2
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 )&=x \left (t \right )-2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=3 x \left (t \right )-4 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.062 (sec). Leaf size: 34

dsolve([diff(x(t),t)=x(t)-2*y(t),diff(y(t),t)=3*x(t)-4*y(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 60

DSolve[{D[x[t],t]==x[t]-2*y[t],D[y[t],t]==3*x[t]-4*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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