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## Homework Statement

The problem is actually the following Separable Differential Equation:

[itex]\frac{dy}{dx}=\frac{xy+3x-y-3}{xy-2x+4y-8}[/itex]

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:

[itex]e^{ln|x|}=x[/itex] ; [itex]ln|xy|=ln|x|+ln|y|[/itex]

## The Attempt at a Solution

[itex]\frac{dy}{dx}=\frac{xy+3x-y-3}{xy-2x+4y-8}[/itex]

[itex](1-\frac{5}{y+3})dy=(1+\frac{5}{x+4})dy[/itex]

[itex]y-5ln|y+3|=x+5ln|x+4|+C[/itex]

[itex]y-x=5(ln|y+3|-ln|x+4|)[/itex]

[itex](1-\frac{5}{y+3})dy=(1+\frac{5}{x+4})dy[/itex]

[itex]y-5ln|y+3|=x+5ln|x+4|+C[/itex]

[itex]y-x=5(ln|y+3|-ln|x+4|)[/itex]

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?

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