Say, we had the following differential equation:
$$\frac{\mathrm{d}y}{\mathrm{d}x}x = 1+y$$
When we re-arrange the equation and integrate both sides, why will the constant of integration ONLY come with the independent variable, $x$?
My assumption is that it's because we can think of a function as having an input and an output. The output is the dependent variable, and input is the independent variable. So, all the work is done on the input, to get a desired output. So, the constant should only come with the independent variable. Am I wrong?
It doesn't matter what side of the equation we put it on. Observe
$$\frac{\mathrm{d}y}{\mathrm{d}x}x=y+1$$
$$\frac{\mathrm{d}y}{y+1}=\frac{\mathrm{d}x}{x}$$
$$\int\frac{\mathrm{d}y}{y+1}=\int\frac{\mathrm{d}x}{x}$$
$$\ln{\left(y+1\right)}+C_y=\ln{x}+C_x$$
Now, here is the reason it doesn't matter: just bring both constants over to either side
$$\ln{\left(y+1\right)}+\left(C_y-C_x\right)=\ln{x}$$
$$\left(y+1\right)\,\mathrm{e}^{C_y-C_x}=x$$
If we want $y$ in terms of $x$, then we can rearrange to get
$$y=x\,\mathrm{e}^{C_x-C_y}-1$$
But, you ask, aren't there two constants now? No, the expression $\mathrm{e}^{C_x-C_y}$ acts as a single constant. If you include an initial condition, then you would solve for this entire expression to get the final solution.