AP® Calculus BC · Unit 7 · Topic 7.7
Finding Particular Solutions Using Initial Conditions and Separation of Variables
A differential equation's general solution is a whole family of curves with a constant C. One known point on the curve lets you solve for C and pin down the single particular solution that passes through it.
Separation of variables gives you a general solution: every value of C traces out a different curve, all satisfying the same equation dy/dx = f(x)g(y). That's an infinite family. An initial condition, a point (x0, y0) the solution must pass through, is one extra piece of information, and one equation can solve for one unknown. Plug the point in right after you integrate, before you do anything else to the equation, so you solve for C while it's still sitting by itself.
The order matters because once you start exponentiating or rearranging to solve for y, C gets buried inside other operations and is much easier to mishandle. Right after integrating you typically have something like ln|y| = (k/2)x2 + C. Substitute the known x and y there, solve the simple equation for C as a number, then substitute that number back in and finish isolating y. Saving the algebra on C for last is the single biggest time-waster in these problems.
This topic is where differential equations stop being abstract families and become concrete answers you can check by plugging back in. It sets up the two growth models that follow: exponential growth and logistic growth are both differential equations solved this exact way, separate variables, integrate, use an initial condition to find C. Everything in those models is this mechanism applied to a specific f(x)g(y).
Key terms
- General solution
- The family of all functions satisfying a differential equation, written with an arbitrary constant C.
- Particular solution
- The single function from that family that also satisfies a given initial condition.
- Initial condition
- A known point (x0, y0) that the solution curve must pass through.
- Separation of variables
- A technique that rewrites dy/dx = f(x)g(y) so all y terms are on one side and all x terms on the other before integrating.
Watch it explained
- How it worksHow do I solve for C right after integrating instead of saving it for the very end?
- Worked exampleFind the particular solution of dy/dx = 2xy with y(0) = 5.
- Worked exampleFind the particular solution of dy/dx = 3x2/(2y) passing through the point (1, 2).
- Common mistakeWhy isn't the domain of a particular solution the whole real line?
Check yourself
End of topic.