Do I need the constant of integration here?

Question

I know for something such as $\int x^2 dx$ would be $\frac{1}{3}x^3 + C_1$ but if you have an equation such as $5y^3 = x^4 $ integrating both sides you would get $ \int 5y^3 dy = \int x^4 dy$, $ \frac{5}{4}y^4 + C_1 = yx^4+ C_2$but my question is: do we need the constants on both sides (are they equal)? I thought they would be the same and cancel out but I'm not sure if I'm correct. Any help would be appreciated. Thanks.

Answer 1

You don't need two arbitrary additive constants. The reason is that if you have an equation of the form $$f(x,y) + C_1 = g(x,y) + C_2,$$ you could rearrange it as $$f(x,y) = g(x,y) + \underbrace{(C_2-C_1)}_{C_3}$$ $$f(x,y)=g(x,y)+C_3$$

Answer 2

They're not in general equal, but all that matters is their difference, so there's only one degree of freedom. You can represent this by writing a constant on just one side. But, as @MPW noted, you can't jum from $5y^3=x^4$ to $\int 5y^3dy=\int x^4dx$, but you could jump from the latter, if it were given (or if e.g. we instead started with $5y^3y^\prime=x^4$), to $\frac54y^4=\frac14x^5+C$.