Differential Equations Exercises

Answer

This looks like it came from using the power rule on the function

y = x⁴.

This is a solution. Since the derivative of a constant is zero, adding a constant to the function x⁴ doesn't change the derivative of that function. So we can write all solutions by

y = x⁴ + C

where C is any constant.

Answer

When we take the derivative of a polynomial the degree drops by 1, so this almost looks like it came from taking the derivative of the function y = x⁶. However, the derivative of x⁶ is 6x⁵. To cancel out the extra 6 we get when taking the derivative, we need to introduce a factor of . A solution to the differential equation could look like this:

All solutions look like this (for any constant C):

Answer

What is a function whose derivative is sin x? One answer is

y = -cos x

(take the derivative of -cos x if you need to convince yourself). To write all solutions, we write

y = -cos x + C

where C is any constant.

Answer

Divide both sides by x to get

y' = 4.

A solution is

y = 4x,

but all solutions can be written as

y = 4x + C

where C is any constant.

Answer

We need to think backwards twice. The first time, if we take the derivative of y' to get -x^-2 we could have started with

Thinking backwards again, what function has as its derivative? This one:

y = ln x.

This is one solution. To find all solutions, start over. What has -x^-2 as its derivative? Any function of the form

where B is any constant. In order to have such a y', we must have started with

y = ln x + Bx + C

where B and C are constants.