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## Homework Statement

Determine whether this is an exact equation. If it is, find the solution.

[itex]\[\frac{\mathrm{d}y }{\mathrm{d} x} = -\frac{ax+by}{bx+cy}\][/itex]

## Homework Equations

## The Attempt at a Solution

If I moved the right side to the left (because an exact equation has the form ZERO on the right side)....

I get [itex]\[\frac{\partial M}{\partial y}(\frac{ax+by}{bx+cy}) \neq 0\][/itex], where as the partial of the second term [itex]\[1\frac{\partial N}{\partial y} = 0\][/itex].

But the book said this is an exact equation.

I attempted the problem first by trying solving it as a homogeneous equation. That is, the equation can be written in the form of the ratio [itex]\[\frac{y}{x}\][/itex]

I tried, and I got the followings

[itex]\[\frac{\mathrm{d}y }{\mathrm{d} x} = -\frac{a+b(y/x)}{b+c(y/x)}\][/itex]

and let v = y/x

[itex]\[v+ x\frac{\mathrm{d}v }{\mathrm{d} x} = -\frac{a+b(v)}{b+c(v)}\][/itex]

I tried to solve this by partial fraction integration but unfortunately i can't because i can't separate the terms since they are written in symbolic forms. I tried to use quadratic formulas but still no help.

So how do I solve this problem?

Thanks.