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Cogito

Differential Equations · Chapter 2 · Lesson 1

Practice Worksheet

Separable Equations

Get the y terms on one side and integrate.

12 problems · about 23 minutes · F-IF.B.6, A-CED.A.1

NameDate

Figure — use these to answer the problems

dy/dx = ky separates immediately and solves to an exponential. Change the rate and watch the curve steepen.
  • The equationdy/dx = a·y
  • Through(0, 1)
k, the rate constant1
starting x0
starting y1
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. y = Ae^(5x) with y = 9 at x = 0. What is A?

    Answer

  2. What makes an equation separable?

    • dy/dx is a function of x times a function of y.
    • dy/dx is a sum of x and y.

  3. y = Ae^(3x) with y = 5 at x = 0. What is A?

    Answer

  4. y = 4e^(2x). What is y at x = 0?

    Answer

  5. Is dy/dx = xy separable? 1 yes, 0 no.

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. Is dy/dx = x + y separable? 1 yes, 0 no.

    Answer

  2. How many arbitrary constants does the general solution of a separable first-order equation have?

    Answer

  3. dy/dx = 3y with y = 2 at x = 0. What is y at x = 0 still?

    Answer

Stretch Yourself

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

4 problems

  1. Order the steps of separation of variables.

    Write 1 to 4 in the boxes to put these in order.

    • Check the equation is separable
    • Move all y terms one side and all x terms the other
    • Integrate both sides and add one constant
    • Use the initial condition to find that constant

  2. Is dy/dx = y² separable? 1 yes, 0 no.

    Answer

  3. The Constant

    y = Ae^(2x) with y = 6 when x = 0. What is A?

    Answer

  4. The Lost One

    dy/dx = xy. Dividing by y risks losing which constant solution?

    Answer