Question

Prove that if `y= sinx/x`, show that `(d^2y)/(dx^2) + 2/x dy/dx + y = 0` ?

Answer 1

See below

Explanation:

If `xy = sin x`, using the product rule:

• `y + xy' = cos x qquad qquad qquad square`

• `y' + y' + x y'' = - sin x qquad triangle`

`triangle/ x + square implies`

`(2y')/x + y'' + color( red)( y + xy') = - color(blue)((sin x)/x) + cos x`

As `color(blue)(y = (sin x)/x)` and `color(red)( y + xy' = cos x)`:

• `(2y')/x + y'' + cos x = - y + cos x`

• `implies y'' + (2y')/x + y = 0`