Differential Equations Workbook For Dummies®

3.2. Getting Answers from Exact Differential Equations

To find the solution to an exact differential equation like this one:

in a systematic way, you need to find a function f(x, y) such that

and

The correct function allows you to write the differential equation like this:

Here's your assignment, should you choose to accept it: Try to integrate M(x, y) and N(x, y) with respect to x and y to see whether you can find f(x, y).

In this case,

M(x, y) = 2xy

which means that

If you integrate both sides, you get

f(x, y) = x²y + g(y)

where g(y) is a function that depends only on y, not on x. What's g(y)? Ah, but you already know the answer to that question because

and

N(x, y) = (1 + x²)

so

Because you know that

f(x, y) = x²y + g(y)

you get this:

So

Integrating with respect ...