Question

Given that y=(k+x)cosx where k is a constant find d^2y\dx^2 +y and show that it's independent of k ?

Answer 1

See below.

Explanation:

`y = (k+x) cos x`

By the Product Rule:

`y' = cos x - (k+x) sin x`

Same again:

`y'' = - sin x - sin x - (k+x) cos x`

`= - 2 sin x - (k+x) cos x`

`implies y'' + y = - 2 sin x`

This is independent of k.

To understand why, consider the simpler: `y = k cos x`.

In this case:
`y' = - k sin x`
`y'' = - k cos x`
`implies y'' + y = 0`

To look at it another way, `y = k cos x` is one of the (complementary/null) solutions to the homogeneous linear DE `y'' + y = 0`. The other is `y = k_2 sin x`. Try that too and see.