One Curve Out of Infinitely Many

You'll learnto turn a general solution into a particular one with the point you were given, to let the sign of that point choose the branch, and to say on what interval the answer is actually a function.

The general solution is a family. An initial condition picks out one member of it, and doing that properly means three things rather than one: solving for the constant, deciding which branch of the answer the point sits on, and saying on what interval the result is defined at all. The first is the step that gets skipped — an answer that still contains the letter C has replied to a question about the equation rather than to the question that was asked. The second and third are the ones that are quietly examined and rarely taught: a square root is two curves, and a formula that blows up somewhere splits into two solutions that no curve joins.

8 STEPS · 6 QUICK CHECKS · SOLVE FOR C, PICK THE BRANCH, NAME THE INTERVAL · v1

the general solution is a family ⇒ y(0) = 3 names one member ⇒ one curve, one branch, one interval

Before you start

What you're looking at

A family of curves, all of which solve the same differential equation, and one marked point. Every curve here is drawn from one function called with a different constant, so the family is genuinely one object.

The question

Which member does the given point name, which branch of it does the sign of that point choose, and how far does that member actually reach?

Watch for

The teal bar under the plot in steps 5 and 6. It is the interval the answer lives on, and its hollow endpoint is where the formula stops having a value — that endpoint is part of the answer, not a footnote to it.

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