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 ?

1 Answer
Feb 15, 2017

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.