2.6. Answers to Separable First Order Differential Equation Problems

Here are the answers to the practice questions I provide throughout this chapter. I walk you through each answer so you can see the problems worked out step by step. Enjoy!

1 Solve this differential equation:

where

y(0) = 5

Solution:

  1. First, multiply both sides by dx:

    dyx3 dx = 0

  2. Separate x and y terms on different sides of the equal sign:

    dy = x3 dx

  3. Then integrate to get

  4. Last but not least, apply the initial condition to find that

    c = 5

  5. So your answer is

2 Figure out the answer to the following:

where

y(0) = 1

Solution: y = sin (x) + 1

  1. Multiply both sides by dx:

    dy − cos (x) dx = 0

  2. Then separate x and y terms on opposite sides of the equal sign:

    dy = cos (x) dx

  3. Next up, integrate:

    y = sin (x) + c

  4. Applying the initial condition tells you that

    c = 1

  5. So the solution is

    y = sin (x) + 1

3 What's the solution to this equation?

where

y(0) = 0

Solution: y = x

  1. First, multiply both sides by dx:

    y dyx dx = 0

  2. Separate x and y terms ...

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