Topic 7.2 Verifying Solutions for Differential Equations: verify that a proposed function satisfies a differential equation by substitution.

What this topic is asking

The College Board (Topic 7.2) asks you to verify that a given function is a solution of a differential equation. You differentiate the proposed function, substitute it (and its derivative) into the equation, and check that both sides are equal as an identity.

The verification method

A worked verification

General versus particular solutions

A differential equation usually has a family of solutions involving an arbitrary constant (the general solution); fixing the constant with an initial condition gives one particular solution. Verification works for both. In the worked example, the constant $C$ cancelled, confirming that the whole family solves the equation. When a specific function is claimed (no free constant) and an initial condition is stated, you verify both that it satisfies the equation and that it passes through the required point. This mirrors the distinction made when solving by separation of variables in the later topics.

Why verification is a separate, scoreable skill

Verification is examined on its own because it tests whether you understand what "solution" means: a function that, together with its derivative, makes the equation an identity. It is also a useful check after solving, since substituting your answer back catches algebra errors. The marks come from showing the substitution into both sides and the simplification, not just asserting the result. The most common error is differentiating incorrectly, especially forgetting the chain rule on an exponential like $e^{-x}$, which then makes the two sides fail to match even for a correct solution. Differentiate carefully, then substitute fully.

Verifying solutions of higher-order equations

Some verification problems involve a second derivative, for example checking that $y = \sin x$ solves $\frac{d^2y}{dx^2} + y = 0$. The method is unchanged: differentiate as many times as the equation requires, substitute $y$ and all its needed derivatives, and confirm the equation holds identically. For $y = \sin x$, $\frac{d^2y}{dx^2} = -\sin x$, and $-\sin x + \sin x = 0$ matches the right side, so it is a solution. The discipline is simply to compute every derivative the equation mentions before substituting, and to keep the chain rule and product rule accurate through the higher derivatives, where errors are easier to make.

Using verification as a confidence check

Beyond being its own question type, verification is the most reliable way to check work after solving a differential equation. Once you have produced a candidate solution by separation of variables, differentiating it and substituting into the original equation confirms whether the algebra was correct. If the two sides do not match, you have caught an error before it costs marks elsewhere. Because differentiation is generally more dependable than the integration and algebra involved in solving, this back-substitution is a cheap and powerful safeguard. Treat it as the final step of any solve, not only as a standalone task.