Verifying Solutions for Differential Equations

A differential equation is a statement about a function and its own derivative, like dy/dx = 2y. Checking whether some proposed y actually works is a different job from finding it in the first place. You don't need to know any method for solving the equation. You only need to differentiate the candidate and see if the relationship holds. That's why verification comes before separation of variables in this unit: it's the simpler skill, and it builds the habit of treating dy/dx as something you compute and compare, not just a symbol.

The mechanism is always the same three moves. Take the candidate function, differentiate it to get dy/dx, then substitute both the function and its derivative into the original equation. If the left side equals the right side for every x, the candidate is a genuine solution. If they only match at special values of x, or never match, it fails. This is just substitution and simplification, the same checking move you use to confirm a root of a polynomial, except now the unknown is a whole function instead of a number.

A solution to a differential equation is never a single number because the equation constrains a relationship between a function and its rate of change, and infinitely many functions can share that relationship, differing by a constant. That's why solutions come in families, like y = Ce^2x for any constant C: each choice of C gives one curve, and together they fill the plane with non-crossing curves all obeying the same slope condition. Verification checks one family member at a time; separation of variables, coming next, is the technique that generates the whole family from scratch.