14.29.4 problem 4

Internal problem ID [2802]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 4. Qualitative theory of differential equations. Section 4.2 (Stability of linear systems). Page 383
Problem number : 4
Date solved : Monday, January 27, 2025 at 06:14:00 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=x \left (t \right )-4 y\\ y^{\prime }&=4 x \left (t \right )-7 y \end{align*}

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 34

dsolve([diff(x(t),t)=1*x(t)-4*y(t),diff(y(t),t)=4*x(t)-7*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-3 t} \left (c_2 t +c_1 \right ) \\ y &= \frac {{\mathrm e}^{-3 t} \left (4 c_2 t +4 c_1 -c_2 \right )}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 46

DSolve[{D[x[t],t]==1*x[t]-4*y[t],D[y[t],t]==4*x[t]-7*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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