separable differential equations with constant I must find all solutions to the following equation: $(1+x^2)y'+ 2xy= 2x$
The answer I got is : $y = 1 + e^c/(1+x^2)$
The correct answer is : $y = 1 + c/(1+x^2)$
My question is, why do I get $e^c$ and not only $c$? Am I missing something?
 A: I'm not sure where you are getting an exponential from. When I try to solve your equation, I get the following:
$$ (1 + x^2) y' + 2xy = ((1 + x^2)y)' = 2x \implies (1 + x^2) y = x^2 + C \implies y = \frac{x^2}{1 + x^2} + \frac{C}{1+x^2}, $$
which seems to be different form the "correct" answer.
Instead, let me show you a different example, where you can more clearly see why you can sometimes get $e^C$. Let's look at the simple equation $y'= y$. Separating the variables gives:
$$ \int \frac{dy}{y} = \int{dx} \implies \ln|y| = x + C \implies |y| = e^C e^x. $$
You can rename the constant $e^C = K$ for convenience. However, there is a small problem. No matter what $C$ is, the new constant $K$ will always be positive. In this case, the absolute value bars compensate for this, since
$$ y = \pm e^C e^x = K e^x. $$
Now, the $\pm$ means that $K$ can be positive and negative (but still not zero). However, setting $K = 0$ gives $y = 0$, which is still a valid solution to the equation (which we originally threw away by dividing by $y$). So, in fact, the new constant $K$ can also be every real number.
The argument I give here works in many situations, and you can usually replace $e^C$ by a new constant $K$ (or just call it $C$ again) without thinking too much about it. But when you're just learning about a new topic, it can help to think about small steps like this.
