### 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:

First, multiply both sides by dx:

dy − x

^{3}dx = 0Separate x and y terms on different sides of the equal sign:

dy = x

^{3}dxThen integrate to get

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

c = 5

So your answer is

2 Figure out the answer to the following:

where

y(0) = 1

Solution: y = sin (x) + 1

Multiply both sides by dx:

dy − cos (x) dx = 0

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

dy = cos (x) dx

Next up, integrate:

y = sin (x) + c

Applying the initial condition tells you that

c = 1

So the solution is

y = sin (x) + 1

3 What's the solution to this equation?

where

y(0) = 0

Solution: y = x

First, multiply both sides by dx:

y dy − x dx = 0

Separate x and y terms ...

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