Consider the differential equation

Question

d²x/dt² - dx/dt - x = cos(t)

This equation could describe a forced damped oscillator, as we will see in Chapter 9. We are told that the differential equation has a solution of the form x(t) = asin(t) + bcos(t). Find a and b.

a = _____

b = _____

Answer 1

x = asint + bcost

dx/dt = acost - bsint

d²x/dt² = -asint - bcost

So, substituting into the differential equation d²x/dt² - dx/dt - x = cost, we have:

(-asint - bcost) - (acost - bsint) - (asint + bcost) = cost

(-a - 2b)cost + (-2a - b)sint = (1)cost + (0)sint

So, -a - 2b = 1 and -2a - b = 0

Multiply the first equation by -2:

2a + 4b = -2

-2a - b = 0

3b = -2

So, b = -2/3 and a = 1/3

x(t) = (1/3)sint - (2/3)cost