Verifying a Solution of a Differential Equation

Whether a solution was reached by variable separation (§4) or was simply given as a candidate function to check, the same direct test confirms whether it genuinely satisfies a differential equation: differentiate the candidate solution, then substitute both the candidate and its derivative(s) back into the original differential equation, and confirm the two sides become identical.

Method:

1. If the candidate solution is given explicitly as $y = f(x)$, differentiate directly. If it is given implicitly as a relation between $x$ and $y$ (e.g. $y^2 = 4ax$), differentiate implicitly with respect to $x$, remembering that $\dfrac{d}{dx}(y^2) = 2y\dfrac{dy}{dx}$ by the chain rule.
2. Substitute the resulting expression for $\dfrac{dy}{dx}$ (and $\dfrac{d^2y}{dx^2}$, if the differential equation is second order) into the differential equation.
3. If any arbitrary constant from the candidate solution still appears after substitution, use the original relation to express that constant in terms of $x$ and $y$, and substitute it out too — the final check should confirm an identity that holds for every $x$, with no leftover constant unless the differential equation itself was stated to still contain one.
4. Confirm the left side and right side of the differential equation match exactly. If they do, the candidate genuinely is a solution; if they do not, either the candidate or the differentiation contains an error.

Illustration: verify that $y=kx$ (for any constant $k$) solves $x\dfrac{dy}{dx}=y$. Differentiating, $\dfrac{dy}{dx}=k$. Substituting into the left side of the differential equation, $x\cdot k = kx$, which is exactly $y$ (the right side) — the identity holds for every $x$ and every $k$, so $y=kx$ is confirmed as the general solution.