Show by differentiating that this function for x(t) does in fact satisfy the differential equation?

We have been told that the differential equation m((d^2x)/(dt^2))=-kx is satisfied by the equation x(t)=x_0cos(sqrt(k/mt)) Show by differentiating that this function for x(t) does in fact satisfy the
differential equation

1 Answer
May 15, 2018

See below.

Explanation:

We are given

[1] =>x(t) = x_0 cos(sqrt(k/m) t)

We take the first derivative:

(dx)/(dt) = x_0(-sin(sqrt(k/m)t)(sqrt(k/m)))

(dx)/(dt) = -x_0 sqrt(k/m) sin(sqrt(k/m)t)

Take the second derivative:

(d^2x)/(dt^2) = -x_0sqrt(k/m) cos(sqrt(k/m)t)(sqrt(k/m))

[2] =>(d^2x)/(dt^2) = -(x_0k)/m cos(sqrt(k/m)t)

Now we use this result and plug it into the differential equation:

m(d^2x)/(dt^2) = -kx

m[-(x_0k)/m cos(sqrt(k/m)t)] = -kx

cancel(m)[cancel(-)(x_0cancel(k))/cancel(m) cos(sqrt(k/m)t)] = cancel(-)cancel(k)x

x_0 cos(sqrt(k/m)t) = x

[3] =>x(t) = x_0 cos(sqrt(k/m)t)

which is precisely what we expect our x(t) to be (notice how by plugging in [2] to the differential equation we recovered [1] in equation [3]). Hence, we have shown that the differential equation produces such a solution.