Question

Find the solution of the system

x′=−4y,y′=2x,

where primes indicate derivatives with respect to t, that satisfies the initial condition x=

Answer 1

To find the solution of the given system of equations x' = -4y and y' = 2x, we differentiate the first equation and substitute the value of y'. This leads to a second-order linear homogeneous differential equation with the solution x(t) = c1 cos(√8t) + c2 sin(√8t), where c1 and c2 are constants.

Step-by-step explanation:

The given system of equations is:

x' = -4y

y' = 2x

To find the solution, we can differentiate the first equation with respect to t:

x'' = -4y'

Substitute the value of y' from the second equation:

x'' = -4(2x)

x'' = -8x

This is a second-order linear homogeneous differential equation. The solution is given by x(t) = c1 cos(√8t) + c2 sin(√8t), where c1 and c2 are constants.