76.7.5 problem 5

Internal problem ID [17478]
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 : 5
Date solved : Tuesday, January 28, 2025 at 10:39:18 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=4 x \left (t \right )-3 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=8 x \left (t \right )-6 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 26

dsolve([diff(x(t),t)=4*x(t)-3*y(t),diff(y(t),t)=8*x(t)-6*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{1} +{\mathrm e}^{-2 t} c_{2} \\ y \left (t \right ) &= 2 \,{\mathrm e}^{-2 t} c_{2} +\frac {4 c_{1}}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 59

DSolve[{D[x[t],t]==4*x[t]-3*y[t],D[y[t],t]==8*x[t]-6*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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