The Answer Is a Function

You'll learnto test a candidate by computing both sides of the equation separately, to see why agreement at one input proves nothing, and to finish an initial value problem by using the point you were given.

Solving an algebraic equation produces a number. Solving a differential equation produces a function, and checking one means doing something you already know how to do. Differentiate the candidate, that is the left side. Substitute the candidate wherever y appears, that is the right side. Then ask whether the two results are the same expression — not the same number at one convenient input, the same expression, for every value of the variable. Two habits carry most of the marks here. Never stop after computing one side, because a single expression has nothing to be compared against. And when a point is supplied, use it: a family with a constant still in it is not the answer to a problem that named a point.

8 STEPS · 6 QUICK CHECKS · BOTH SIDES, EVERY INPUT, EVERY GIVEN FACT · v1

dy/dx = 2y ⇒ left side dy/dx ⇒ right side 2y ⇒ same expression, for every x

Before you start

What you're looking at

Two boxes, one for each side of the equation. The left box differentiates the candidate; the right box substitutes it. A teal line underneath says whether the two agree.

The question

Given an equation and a candidate function, do the two sides come out as the same expression — and if a point was supplied as well, does the candidate pass through it?

Watch for

The step where a candidate matches at exactly one input and fails everywhere else. That is what a check at a single value of x cannot see.

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