Question

Find all values of k for which the function y=sin(kt) satisfies the differential equation y′′+16y=0. Separate your answers by commas. isn't the answer just ±4?

Answer 1

0, 4, -4 and they may want you to mention formally all the kt multiples of
`\pi`.

Explanation:

Let's do the second derivative of the function:
`y(t)=sin(k\,t)`

`y'(t) =k\,cos(k\,t)\\y`

So now we want:

`y`

Then we have to include the zeros of the binomial (
`16-k^2`) which as you say are +4 and -4, and also the zeros of
`sin(kt)`, which include all those values of

`kt=0\,,\,\pi\,\,,\,2\pi\, ,\,etc.`

So an extra one that they may want you to include is k = 0