Separable Differential Equations

An equation whose unknown is a function.

11 problems · about 24 minutes · AP Calculus AB 7.6, 7.7

NameDate

Figure — use these to answer the problems

Exponential growth is the solution of dy/dx = ky.

y = 1 · 2^x + 0

Initial amount1

Growth rate2

Figure 1

Straightforward practice. Get the method working first.

4 problems

dy/dx = 2x with y = 9 at x = 0. What is y at x = 2?
Answer
dy/dx = 4x with y = 1 at x = 0. What is C?
Answer
dy/dx = 3 with y = 2 at x = 0. What is y at x = 4?
Answer
Does a differential equation have one solution or a family? 1 for one, 2 for a family.
Answer

The same ideas with more to keep track of.

3 problems

5
Order the steps for solving a separable equation with an initial condition.
Write 1 to 4 in the boxes to put these in order.
- Separate the variables
- Integrate both sides
- Add the constant of integration
- Apply the initial condition
dy/dx = 2x with y = 0 at x = 2. What is C?
Answer
Exponential growth solves dy/dx = ky. Is the rate proportional to the amount? 1 for yes, 0 for no.
Answer

Mixed problems. Work out what kind of question it is before you start.

4 problems

dy/dx = 6x² with y = 1 at x = 0. What is y at x = 1?
Answer
Forgetting + C loses how many solutions? 1 for one, 2 for infinitely many.
Answer
The Constant
dy/dx = 2x with y = 7 at x = 0. What is C?
Answer
Evaluating
With y = x² + 7, what is y at x = 3?
Answer