Question

Verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation: $y=\sqrt{1+x^2}\::y'=\frac{xy}{1+x^2}$

Answer 1

$y=\sqrt{1+x^2}$

Differentiating both sides of the equation with respect to x, we get:

$y'=\frac{d}{dx}(\sqrt{1+x^2})$

$y'=\frac{1}{2 \sqrt{1+x^2}}. \frac{d}{dx}(1+x^2)$

$y'=\frac{2x}{2\sqrt{1+x^2}}$

$y'=\frac{x}{\sqrt{1+x^2}}$

$\Rightarrow y' = \frac{x}{1+x^2} *\sqrt{1+x^2}$

$\Rightarrow y'=\frac{x}{1+x^2}.y$

$\Rightarrow y'=\frac{xy}{1+x^2}$

∴L.H.S.=R.H.S.

Hence, the given function is the solution of the corresponding differential equation.