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Cogito

Differential Equations · Chapter 3 · Lesson 1

Practice Worksheet

Linear First-Order Equations

When separation fails, multiply by something clever.

12 problems · about 24 minutes · F-IF.B.6, A-CED.A.1

NameDate

Figure — use these to answer the problems

This field belongs to a linear equation that does not separate. Follow solutions from several starts and see how they converge.
  • The equationdy/dx = a·(x + y)
  • Through(-2, 2)
starting x-2
starting y2
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. dy/dx + 6y = 18. What constant value does y settle toward?

    Answer

  2. What does the integrating factor accomplish?

    • It turns the left side into a single product-rule derivative.
    • It cancels the right-hand side.

  3. dy/dx + 4y = 8. What is P?

    Answer

  4. dy/dx + 4y = 8. What constant value does y settle toward?

    Answer

  5. dy/dx + 3y = 12. What constant value does y settle toward?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. Is dy/dx + 2y = x linear? 1 yes, 0 no.

    Answer

  2. Is dy/dx + y² = x linear? 1 yes, 0 no.

    Answer

  3. y = 7 + Ce^(−3x) with y = 10 at x = 0. What is C?

    Answer

Stretch Yourself

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

4 problems

  1. Order the steps of the integrating factor method.

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

    • Write the equation in standard form
    • Compute the integrating factor from P
    • Multiply through and recognise the product rule
    • Integrate both sides and divide the factor back out

  2. y = 5 + Ce^(−x). What value does y approach as x grows large?

    Answer

  3. The Factor

    dy/dx + 5y = 1. What is the coefficient P?

    Answer

  4. The Long Run

    y = 3 + Ce^(−2x). What value does y approach as x grows large?

    Answer