변수분리형 미분방정식

To say $y=f(x)+C$ is a solution to the differential equation means:

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}x}\{f(x)+C\}&=x(f(x)+C) \\ \text{Since $y=f(x)$ is a solution, we know $\frac{\mathrm{d}}{\mathrm{d}x}\{f(x)\}=xf(x)$. Also, $\frac{\mathrm{d}}{\mathrm{d}x}\{f(x)+C\}=\frac{\mathrm{d}}{\mathrm{d}x}\{f(x)\}$. So, $\frac{\mathrm{d}}{\mathrm{d}x}\{f(x)+C\}=xf(x)$.} \\ xf(x)&=x(f(x)+C)\\ 0&=xC \end{aligned}$$

Our equation should hold for all $x$ in our domain, and for the derivative to $y$ with respect to $x$ to make sense, our domain should not be a single point. So, there is some $x$ in our domain such that $x \neq 0$. Therefore, the $C$ must be zero. So, $f(x)+C$ is not a solution to the differential equation for any constant $C$.

When we're finding a general antiderivative, we add “$+C$" at the end. When we're finding a general solution to a differential equation, the “$+C$" gets added when we antidifferentiate–we don't add another one at the end of our work.