Q.Verify that $y^2 = 4ax$, where $a$ is an arbitrary constant, satisfies the differential equation $2x\dfrac{dy}{dx} = y$ (i.e., show the constant $a$ has been correctly eliminated).

Differentiate implicitly, then use the original relation to eliminate the constant $a$ from the result.

The candidate relation is $y^2 = 4ax$, with $a$ an arbitrary constant.

Differentiating implicitly with respect to $x$ (using the chain rule on $y^2$): $2y\dfrac{dy}{dx} = 4a$, so $\dfrac{dy}{dx} = \dfrac{2a}{y}$.

This still contains the constant $a$, so — exactly as in the formation method of §3 — express $a$ from the original relation: $a = \dfrac{y^2}{4x}$.

Substituting: $\dfrac{dy}{dx} = \dfrac{2\left(\dfrac{y^2}{4x}\right)}{y} = \dfrac{y^2}{2xy} = \dfrac{y}{2x}$.

Rearranging: $2x\dfrac{dy}{dx} = y$ — exactly the differential equation to be verified, now completely free of $a$, confirming that $y^2=4ax$ is indeed a solution. …

A common error is stopping at $\dfrac{dy}{dx}=\dfrac{2a}{y}$ and declaring the verification complete — but this still contains the constant $a$, while the differential equation being verified, $2x\dfrac{dy}{dx}=y$, contains no constant at all; whenever a constant survives after the f …