I've always been curious about where the +C comes from. I fully understand its use in calculus, but recently I've been studying physics and realized that integrating time we tend to consider the lower bound as 0 and define the respective lower bound for the dependent variable as $y_0$. I am curious if my math work below is accurate for representing +C $$dy=dx$$ $$\int_{y_0}^y{dy}=\int_0^xdx$$ $$y-y_0=x$$ $$y=x+y_0$$
I will give this work done in a real example.
$$\frac{dy}{y}=6x^2dx$$ find solution with initial conditions y=0 x=4 $$\int_{y_0}^y\frac{dy}{y}=\int_0^x 6x^2dx$$ $$\ln|y||_{y_0}^y=2x^3|_0^s$$ $$\ln|y|-\ln|y_0|=2x^3-2(0)^3$$ $$\ln|y|=2x^3+\ln|y_0|$$ $$e^{\ln|y|}=e^{2x^3+\ln|y_0|}$$ $$y=y_0e^{2x^3}$$ $$y=4e^{2x^3}$$
My main question for this arose as integrating the dy side didn't give a +C but when we integrate the dx side that side gets a +C. I wonder if my math work where we consider the lower bound of the dependent variable as 0 to be the reason that +C only shows on the dx side. And because we define y as dependent on the x value, we simply consider the respective lower bound as $y_0$ which then gets eventually moved to the x side as +C.
My one exception I’ve found is the case of functions where having the lower bound as 0 for x returns two values. For example $e^x$ which integrated from 0 to x becomes $e^x-1$
