76.10.6 problem 6

Internal problem ID [17528]
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 : 6
Date solved : Tuesday, January 28, 2025 at 10:39:58 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-8 x \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = 2 \end{align*}

✓ Solution by Maple

Time used: 0.062 (sec). Leaf size: 29

dsolve([diff(x(t),t) = 2*y(t), diff(y(t),t) = -8*x(t), x(0) = 1, y(0) = 2], singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 30

DSolve[{D[x[t],t]==2*y[t],D[y[t],t]==-8*x[t]},{x[0]==1,y[0]==2},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \sin (4 t)+\cos (4 t) \\ y(t)\to 2 (\cos (4 t)-\sin (4 t)) \\ \end{align*}