# Determine whether this is an exact equation [differential]

## Homework Statement

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

$$\frac{\mathrm{d}y }{\mathrm{d} x} = -\frac{ax+by}{bx+cy}$$

## 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 $$\frac{\partial M}{\partial y}(\frac{ax+by}{bx+cy}) \neq 0$$, where as the partial of the second term $$1\frac{\partial N}{\partial y} = 0$$.

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 $$\frac{y}{x}$$

I tried, and I got the followings
$$\frac{\mathrm{d}y }{\mathrm{d} x} = -\frac{a+b(y/x)}{b+c(y/x)}$$
and let v = y/x
$$v+ x\frac{\mathrm{d}v }{\mathrm{d} x} = -\frac{a+b(v)}{b+c(v)}$$

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.

lurflurf
Homework Helper
An exact differential looks like
dU=Ux dx + Uy dy
Where Ux and Uy are partial derivative with respect to x and y respectively
Notice that under mild conditions on Ux and Uy we have
Uxy=Uyx
by equality of mixed partial derivatives
Write the equation in the form
F dx +G dy = 0
If this is an exact differential
Fy=Gx
so check if it is
An exact differential equation such as
dU=0
is simple to solve
U=some constant.

Last edited:
OH. the partial of the two gives b
the form M(x,y) = N(x,y) y' = 0 can be written as F dx +G dy = 0 if we multiply both side by dx, right?

the dU is always 0 if it has to be an exact, am i correct?

lurflurf
Homework Helper
That is right. Now try to find U by integration.