Question

If $y\sqrt{x^2 + 1} = \log\left(\sqrt{x^2 + 1} - x\right)$, show that: \[ (x^2 + 1)\frac{dy}{dx} + xy + 1 = 0 \]

Hint:

Recognizing that $\frac{d}{dx}\left[\log(\sqrt{x^2+1}-x)\right] = -\frac{1}{\sqrt{x^2+1}}$ is a standard identity saves time. This expression is closely related to the derivative of the inverse hyperbolic sine function!

Answer 1