76.10.6 problem 6

Internal problem ID [17449]
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 : Thursday, March 13, 2025 at 10:08:34 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*}

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

ode:=[diff(x(t),t) = 2*y(t), diff(y(t),t) = -8*x(t)]; 
ic:=x(0) = 1y(0) = 2; 
dsolve([ode,ic]);
 

\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*}

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

ode={D[x[t],t]==2*y[t],D[y[t],t]==-8*x[t]}; 
ic={x[0]==1,y[0]==2}; 
DSolve[{ode,ic},{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*}

✓ Sympy. Time used: 0.069 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*y(t) + Derivative(x(t), t),0),Eq(8*x(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {C_{1} \sin {\left (4 t \right )}}{2} + \frac {C_{2} \cos {\left (4 t \right )}}{2}, \ y{\left (t \right )} = C_{1} \cos {\left (4 t \right )} - C_{2} \sin {\left (4 t \right )}\right ] \]