Concept preview

What you will be able to do

1. 01Verify that a proposed function satisfies a differential equation by differentiating it and substituting $y$ and $y'$ (and $y''$ when needed) to confirm both sides agree for all $x$.
2. 02Check whether a solution meets a given initial condition, and use that condition to select a particular solution from a general family that contains an arbitrary constant $C$.
3. 03Differentiate candidate solutions correctly — especially exponentials via the chain rule — and explain the difference between a general solution and a particular solution.

A function is a solution of a differential equation if substituting it makes the equation true for every $x$ in the domain. To verify a candidate $y$, you carry out three steps. First, differentiate the candidate to find $y'$ (and $y''$ if the equation contains a second derivative). Second, substitute both $y$ and its derivative into the equation. Third, simplify each side and check whether the left side equals the right side identically. If they match for all $x$, the candidate is a solution; if they differ, it is not.