# Playing With Logs while solving y-x=5(ln|y+3|-ln|x+4|) for y.

## Homework Statement

The problem is actually the following Separable Differential Equation:
$\frac{dy}{dx}=\frac{xy+3x-y-3}{xy-2x+4y-8}$​
I am required to find y(x).

2. Homework Equations and techniques

- factorization (applied on the numerator and the denominator in the problem equation)
- basic integration (applied after separating the variables)
- at least the following logarithmic identities:
$e^{ln|x|}=x$ ; $ln|xy|=ln|x|+ln|y|$​

## The Attempt at a Solution

$\frac{dy}{dx}=\frac{xy+3x-y-3}{xy-2x+4y-8}$

$(1-\frac{5}{y+3})dy=(1+\frac{5}{x+4})dy$

$y-5ln|y+3|=x+5ln|x+4|+C$

$y-x=5(ln|y+3|-ln|x+4|)$

I also put this result in Wolfram Alpha, but it could not solve the equation for y. My instructor apparently believes that an y=f(x) can be obtained. Is there any way to do so?

Last edited:

## Answers and Replies

haruspex
Science Advisor
Homework Helper
Gold Member
2020 Award
You appear to have a sign wrong in the middle two equations, but since it comes out right at the end maybe this was a copying out error.
i don't see a way to get it into y = f(x) form either.

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