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Cogito

Differential Equations · Chapter 1 · Lesson 1

Practice Worksheet

What a Differential Equation Is

An equation whose unknown is a whole function.

12 problems · about 22 minutes · F-IF.B.6, A-CED.A.1

NameDate

Figure — use these to answer the problems

Every little dash shows the slope the solution must have there. Release a point and watch a solution curve follow them.
  • The equationdy/dx = a·y
  • Through(0, 1)
the growth rate1
starting x0
starting y1
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. y″ + 4y′ + 3y = 0. What is the order?

    Answer

  2. What is the unknown in a differential equation?

    • A whole function, described by how it changes.
    • A single number.

  3. dy/dx = x². What is the order?

    Answer

  4. y‴ − y = 0. What is the order?

    Answer

  5. dy/dt = 3y with y = 7. What is dy/dt?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. dy/dx = 2y and y = 0. What is dy/dx?

    Answer

  2. Is y = 5 a solution of dy/dx = 0? 1 yes, 0 no.

    Answer

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

    Answer

Stretch Yourself

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

4 problems

  1. Sort each equation by its order.

    Write each item under the heading it belongs to: dy/dx = 4y · y″ + y = 0 · dy/dt + 2y = t · y″ − 3y′ = 5

    First order

    Second order

  2. An initial value problem for a first-order equation. How many solutions does it usually have?

    Answer

  3. The Order

    y″ + 3y′ − y = 0. What is the order of this equation?

    Answer

  4. The Rate

    dy/dt = 5y and y = 4 at the moment asked. What is dy/dt then?

    Answer