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## Homework Statement

find the general solution using integrating factors.

xy' - 2y = 3x

## Homework Equations

## The Attempt at a Solution

x(dy/dx) = 3x + 2y

x*dy = (3x + 2y)dx

(-3x + 2y)dx + xdy = 0

My = -3x + 2

Nx = 1

Not Exact (hence the use of integrating factors)

μ(x)(-3x + 2y)dx + μ(x)xdy = 0

Differentiating with respect to y for M and x for N

μ(x)(-3x + 2) = μ'(x)x + μ(x)

trying to simplify

-3xμ(x) + 2μ(x) - μ(x) = μ'(x)

-3xμ(x) - μ(x) = μ'(x)

-μ(x)(3x - 1) = μ'(x)

I know I need to solve for μ(x) to multiply M and N by, I am not sure if I have done everything correct up until now. I assume that I possibly just need a little help with the algebra to be able to set up an integral to solve for μ(x) but I am not sure.

Any help would be greatly appreciated, I'm not sure if I missed something from the beginning or am almost there. Thanks ahead of time!

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