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I have the following differential equation:

$x-y = \frac{dy}{dx}$

I'm trying to solve for $y$, is this even possible?

I've tried doing the following:

$x-y = \frac{dy}{dx}$

$\int (x-y) dx = \int dy$

$\frac{x^2}{2} -xy = y$ (let $C = 0$)

Then I substitude $x$ for a value and solve for $y$.

I'm unsure whether this can be done or not since I'm treating $y$ as a constant when I'm integrating when in fact it isn't.

Any help is appreciated.

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Note that unlike other functions you have possibly met where you can do partial differentiation, $y=f(x)$ is dependent on $x$ in this case.

As the LHS involves a subtraction sign, not multiplication, you cannot use separation of variables.

Rearranging the equation, you get $y'+y=x$ so $P=Q=1$. Then $I=\exp(\int P\,dx)$ and use the integrating factor method to solve $yQ=\int IQ\,dx$.

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No, you can not treat $y$ as a constant on the left side, as it is a function depending on $x$. You can rewrite the equation as $$ y'(x)+y(x)=x, $$ which perhaps helps to associate the correct solution method.


Or set $u=y-x$, then $u'=y'-1=x-y-1=-u-1$, in the next iteration set $u=y-x+1$, then $u'=y'-1=x-y-1=-u$ and you should know how to solve $u'=-u$.

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