Cogito
AP Calculus AB · Chapter 7 · Lesson 1
Separable Differential Equations
An equation whose unknown is a function.
11 problems · about 24 minutes · AP Calculus AB 7.6, 7.7
What this lesson teaches
The student solves a separable differential equation and applies an initial condition.
- A differential equation relates a function to its derivative.
- Separate the variables, integrate both sides, and keep the constant.
- An initial condition selects one member from the family of solutions.
Warm Up
Straightforward practice. Get the method working first.
4 problemsdy/dx = 2x with y = 9 at x = 0. What is y at x = 2?
Answer 13
Why 13.
dy/dx = 4x with y = 1 at x = 0. What is C?
Answer 1
Why Substitute zero.
dy/dx = 3 with y = 2 at x = 0. What is y at x = 4?
Answer 14
Why y = 3x + 2.
Does a differential equation have one solution or a family? 1 for one, 2 for a family.
Answer 2
Why Until a condition is applied.
Build It Up
The same ideas with more to keep track of.
3 problemsOrder the steps for solving a separable equation with an initial condition.
Answer 1. Separate the variables 2. Integrate both sides 3. Add the constant of integration 4. Apply the initial condition
Why The condition comes last.
dy/dx = 2x with y = 0 at x = 2. What is C?
Answer -4
Why 0 = 4 + C.
Exponential growth solves dy/dx = ky. Is the rate proportional to the amount? 1 for yes, 0 for no.
Answer 1
Why Read the equation aloud.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsdy/dx = 6x² with y = 1 at x = 0. What is y at x = 1?
Answer 3
Why y = 2x³ + 1.
Forgetting + C loses how many solutions? 1 for one, 2 for infinitely many.
Answer 2
Why C ranges over the reals.
The Constant: dy/dx = 2x with y = 7 at x = 0. What is C?
Answer 7
Why C = 7.
Evaluating: With y = x² + 7, what is y at x = 3?
Answer 16
Why 16.