Question

Differential equation representing the family of curves y = e^x (Acosx + Bsinx) is `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + 2y` = 0

पर्याय

• True

• False

Answer 1

This statement is True.

Explanation:

Given equation is y = e^x (Acosx + Bsinx) 

Differentiating both sides, we get

`("d"y)/("d"x)` = e^x (–A sin x + B cos x) + (A cos x + B sin x) e^x

`("d"y)/("d"x)` = e^x (–A sin x + B cos x) + y

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

`("d"^2y)/("d"x^2) = "e"^x (-"A" cosx - "B" sinx) + (-"A" sinx + "B"cosx) . "e"^x + ("d"y)/("d"x)`

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

`("d"^2y)/("d"x^2) = - y + y + 2("d"y)/("d"x)`

∴ `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + 2y` = 0