76.10.1 problem 1

Internal problem ID [17523]
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 3.6 (A brief introduction to nonlinear systems). Problems at page 195
Problem number : 1
Date solved : Tuesday, January 28, 2025 at 10:39:54 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right ) = 4\\ y \left (0\right ) = 2 \end{align*}

✓ Solution by Maple

Time used: 0.061 (sec). Leaf size: 19

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

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

✓ Solution by Mathematica

Time used: 0.040 (sec). Leaf size: 22

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

\begin{align*} x(t)\to 4 e^{-t} \\ y(t)\to 2 e^{-2 t} \\ \end{align*}