Question

If y = 5 cos x – 3 sin x, prove that `(d^2y)/(dx^2) + y = 0`

Answer 1

Given, y = 5 cos x - 3 sin x

Differentiating both sides with respect to x,

`dy/dx = 5  d/dx cos x - 3  d/dx  sin x`

= 5 (- sin x) - 3 cos x = - 5 sin x - 3 cos x

Differentiating both sides again with respect to x,

`(d^2 y)/dx= - 5  d/dx  sin x - 3  d/dx  cos x`

= - 5 cos x - 3 (- sin x) = 3 sin x - 5 cos x

Hence, `(d^2 y)/dx^2 + y ` = 3 sin x - 5 cos x + 5 cos x - 3 sin x     ...(On substituting the value of y)

= 0