Luca B. | Certified Educator

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You should isolate the variables such that:

`x(dy/dx)=3-y => (dx)/x = (dy)/(3-y)`

Integrating both sides yields:

`int (dy)/(3-y) = int (dx)/x`

You should use substitution to evaluate `int (dy)/(3-y)`  such that:

`3-y = t => -dy = dt`

`int (dy)/(3-y) = int -(dt)/t = -ln|t| + c`

`int (dy)/(3-y) = -ln|3-y| + c`

`-ln|3-y| = ln|x| + c`

You need to find the constant c, hence, you need to substitute 1 for x and 1 for y in `-ln|3-y| = ln|x| + c`  such that:

`-ln(3-1) = ln1 + c`

Since `ln 1 = 0 => c = -ln 2 = ln(1/2)`

`-ln|3-y| = ln|x| + ln(1/2) => ln(1/(3-y)) = ln|x| + ln(1/2)`

Using logarithmic identities yields:

`ln(1/(3-y)) = ln (x/2) => 1/(3-y) = x/2 => 2 = x(3-y)`

`2 = 3x - xy => 2 - 3x = -xy => y = (3x-2)/x`

Hence, evaluating the solution to the given equation, under the given conditions, yields `y = (3x-2)/x` .

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