76.7.16 problem 16

Internal problem ID [17489]
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 : 16
Date solved : Tuesday, January 28, 2025 at 10:39:26 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = 3 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 33

dsolve([diff(x(t),t) = -2*x(t)+y(t), diff(y(t),t) = -5*x(t)+4*y(t), x(0) = 1, y(0) = 3], singsol=all)
 

\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{3 t}}{2}+\frac {{\mathrm e}^{-t}}{2} \\ y \left (t \right ) &= \frac {5 \,{\mathrm e}^{3 t}}{2}+\frac {{\mathrm e}^{-t}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 42

DSolve[{D[x[t],t]==-2*x[t]+y[t],D[y[t],t]==-5*x[t]+4*y[t]},{x[0]==1,y[0]==3},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{2} e^{-t} \left (e^{4 t}+1\right ) \\ y(t)\to \frac {1}{2} e^{-t} \left (5 e^{4 t}+1\right ) \\ \end{align*}