Cogito
Differential Equations · Chapter 2 · Lesson 1
Separable Equations
Get the y terms on one side and integrate.
12 problems · about 23 minutes · F-IF.B.6, A-CED.A.1
What this lesson teaches
The student solves separable differential equations and applies initial conditions.
- A separable equation splits into a function of x times a function of y.
- Move each variable to its own side, integrate, and keep a single constant.
- Dividing by a function of y can lose the constant solution where it vanishes.
Warm Up
Straightforward practice. Get the method working first.
5 problemsy = Ae^(5x) with y = 9 at x = 0. What is A?
Answer 9
Why 9.
What makes an equation separable?
Answer dy/dx is a function of x times a function of y.
Why A product of the two.
y = Ae^(3x) with y = 5 at x = 0. What is A?
Answer 5
Why Substitute x = 0.
y = 4e^(2x). What is y at x = 0?
Answer 4
Why e to the zero.
Is dy/dx = xy separable? 1 yes, 0 no.
Answer 1
Why A function of x times a function of y.
Build It Up
The same ideas with more to keep track of.
3 problemsIs dy/dx = x + y separable? 1 yes, 0 no.
Answer 0
Why A sum cannot be pulled apart this way.
How many arbitrary constants does the general solution of a separable first-order equation have?
Answer 1
Why One integration each way, one surviving constant.
dy/dx = 3y with y = 2 at x = 0. What is y at x = 0 still?
Answer 2
Why The initial condition itself.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsOrder the steps of separation of variables.
Answer 1. Check the equation is separable 2. Move all y terms one side and all x terms the other 3. Integrate both sides and add one constant 4. Use the initial condition to find that constant
Why The constant cannot be found before it exists.
Is dy/dx = y² separable? 1 yes, 0 no.
Answer 1
Why A function of y alone still counts.
The Constant: y = Ae^(2x) with y = 6 when x = 0. What is A?
Answer 6
Why 6.
The Lost One: dy/dx = xy. Dividing by y risks losing which constant solution?
Answer 0
Why y = 0.