Question

For the following differential equation verify that the accompanying function is a solution:

Differential equation	Function
\[x + y\frac{dy}{dx} = 0\]	\[y = \pm \sqrt{a^2 - x^2}\]

Answer 1

We have,
\[y = \pm \sqrt{a^2 - x^2}\]
\[ \Rightarrow y^2 = a^2 - x^2 . . . . . \left( 1 \right)\]
Given differential equation:
\[x + y\frac{dy}{dx} = 0\]
Differentiating both sides of (1) with respect to x, we get
\[2y \frac{dy}{dx} = - 2x\]
\[ \Rightarrow y \frac{dy}{dx} = - x\]
\[ \Rightarrow x + y \frac{dy}{dx} = 0\]
Hence, the given function is the solution to the given differential equation.