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
NJ
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.

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