Question

Verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:$y=\sqrt{a^2-x^2}\,\,\, x∈(-a,a) :x+y (\frac{dy}{dx}) =0(y≠0)$

Answer 1

$y=\sqrt{a^2-x^2}$
Differentiating both sides of this equation with respect to x,we get:
$\frac{dy}{dx}=\frac{d}{dx}(√a^2-x^2)$
$⇒\frac{dy}{dx}=\frac{1}{2√a^2-x^2}.\frac{d}{dx}(a^2-x^2)$
$=\frac{1}{2√a^2-x^2}(-2x)$
$=-\frac{x}{√a^2-x^2}$
Substituting the value of $\frac{dy}{dx}$ in the given differential equation,we get:
L.H.S.=$x+y\frac{dy}{dx}=x+√a^2-x^2×\frac{-x}{√a^2-x^2}$
$=x-x$
$=0$
=R.H.S.
Hence,the given function is the solution of the corresponding differential equation.