# Equation of the type mdx+ndy=0

## Homework Statement

Solve:

$$(3x^4siny-y^3)dx+(x^5cosy+3xy^2)dy=0$$

## The Attempt at a Solution

At first I thought it was a simple equation of the type mdx+ndy=0, but when I integrated m wrt x and ignored all terms containing x in n (all of them in this case), and added, I didnt get the solution.

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## Answers and Replies

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Dick
Science Advisor
Homework Helper
The problem looks suspiciously like it was intended to be an exact equation but something messed up in transcribing it. Because it's not.

Yes...it does look like it would be an exact equation, but it's not quite there.

Is there anything we can do to make it so?

how about finding an integrating factor to multiply through in order to convert it to exact?

if...

$$\frac{My-Nx}{N}$$

is a function of x only, then the solution to:

$$\frac{d\mu}{dx} = \frac{My-Nx}{N} \mu$$

gives $$\mu$$ as the appropriate integrating factor

or if

$$\frac{Nx-My}{M}$$

is a function of y only, then the solution to:

$$\frac{d\mu}{dx} = \frac{Nx-My}{M} \mu$$

gives you the integrating factor, $$\mu$$

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Dick
Science Advisor
Homework Helper
Sure enough. There IS an integrating factor. Can you find it, chaoseverlasting?

No man. I have no idea what you guys are doing. The only integrating factor I know of is in the Linear DE. I dont think I can reduce this to a LDE

HallsofIvy
Science Advisor
Homework Helper
Then I strongly suggest you go back to your textbook and review "exact equations" and "integrating factors"!

The thing is, Im not even in college yet. I have a basic understanding of what DE's are but this one came in an exam. We havent done this sort of thing.

JasonRox
Homework Helper
Gold Member
The thing is, Im not even in college yet. I have a basic understanding of what DE's are but this one came in an exam. We havent done this sort of thing.
We believe you.

Dick
Science Advisor
Homework Helper
Well, then just try and multiply the equation by x^n. Then apply the exactness test (dM/dy=dN/dx) and determine an n that works. Then go back and try your integration again. That's an example of an integrating factor.