Question

The differential equation for y = Acos αx + Bsin αx, where A and B are arbitrary constants is ______.

Options

• `("d"^2y)/("d"x^2) - alpha^2y` = 0

• `("d"^2y)/("d"x^2) + alpha^2y` = 0

• `("d"^2y)/("d"x^2) + alphay` = 0

• `("d"^2y)/("d"x^2) - alphay` = 0

Answer 1

The differential equation for y = Acos αx + Bsin αx, where A and B are arbitrary constants is `("d"^2y)/("d"x^2) + alpha^2y` = 0.

Explanation:

Given equation is : y = A cos a x + B sin a x

Differentiating both sides w.r.t. x, we have

`("d"y)/("d"x) = -"A" sin alpha x * alpha + "B" cos alpha x * alpha`

= `- "A" alpha sin alphax + "B" alpha cos alpha x`

Again differentiating w.r.t. x, we get

`("d"^2y)/("d"x^2) = -"A"alpha^2 cos alpha x - "B" alpha^2 sin alpha x`

⇒ `("d"^2y)/("d"x^2) = -alpha^2 ("A" cos alphax + "B" sin alpha x)`

⇒ `("d"^2y)/("d"x^2) = - alpha^2y`

⇒ `("d"^2y)/("d"x^2) + alpha^2y` = 0