Suppose we have a separable differential equation and the last form of the equation is:
$$3y^2 \, \mathrm dy = x \, \mathrm dx$$
After solving the equation, why do we write $+C$ to only right-hand side of the equation and not the left one? As far as I can see solution set is changing depends on it.
Both integrations yield constants. Technically the integration should look like:
$$\int3y^2\, \text{d}y = \int x\,\text{d}x \\ y^3 + C_1 = \frac{1}{2}x^2 + C_2 $$
To simplify, we just move the constants together, by subtracting $C_1$ from both sides, yielding:
$$y^3 = \frac{1}{2}x^2 + C_2 - C_1$$
Since $C_1$ and $C_2$ are both unknown constants, their subtraction will also result in an unknown constant. So we simply set $C = C_2 - C_1$, and get:
$$y^3 = \frac{1}{2}x^2 + C$$
Remember, $C$ is an unknown constant, and can be positive or negative. Therefore, putting $+C$ on the right-hand side is equivalent to putting $+C$ on the left-hand side. It just changes what the constant is, which doesn't matter, since we don't know it anyway. If it were on the left, we would just change the definition of $C$ to be $C = C_1 - C_2$. Again, it doesn't matter because it is unknown. As long as there is a single definition for it, you can put it on whatever side you want.