74.1.61 problem 83

Internal problem ID [15841]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 1. Introduction to Differential Equations. Exercises 1.1, page 10
Problem number : 83
Date solved : Tuesday, January 28, 2025 at 08:09:16 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-5 x \left (t \right )+4 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )+2 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 4\\ y \left (0\right ) = 0 \end{align*}

✓ Solution by Maple

Time used: 0.053 (sec). Leaf size: 33

dsolve([diff(x(t),t) = -5*x(t)+4*y(t), diff(y(t),t) = 2*x(t)+2*y(t), x(0) = 4, y(0) = 0], singsol=all)
 

\begin{align*} x \left (t \right ) &= \frac {4 \,{\mathrm e}^{3 t}}{9}+\frac {32 \,{\mathrm e}^{-6 t}}{9} \\ y \left (t \right ) &= \frac {8 \,{\mathrm e}^{3 t}}{9}-\frac {8 \,{\mathrm e}^{-6 t}}{9} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 40

DSolve[{D[x[t],t]==-5*x[t]+4*y[t],D[y[t],t]==2*x[t]+2*y[t]},{x[0]==4,y[0]==0},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {4}{9} e^{-6 t} \left (e^{9 t}+8\right ) \\ y(t)\to \frac {8}{9} e^{-6 t} \left (e^{9 t}-1\right ) \\ \end{align*}