Question

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:

`y = sqrt(a^2 - x^2 )  x in (-a,a) : x + y  dy/dx = 0(y != 0)`

Answer 1

We have, `y = sqrt (a^2 - x^2)`                      ...(i)

Differentiating (I) w.r.t. x, we get

`y' = (1xx (-2x))/(2sqrt(a^2 - x^2))`

⇒ `y' = (-x)/sqrt (a^2 - x^2)`

⇒ `y' = (-x)/y`             (Using (i))

⇒ yy' = -x

⇒ x + yy' = 0

∴ `y = sqrt (a^2 - x^2)` is a solution of the given differential equation.