Why does substitution yield a different result to integration? (Kinematics Example)

Question

Method 1:

Let $x=t^2$, find a general solution of $v$ in terms of $x$. (Kinematics, $v=x'(t)$)

$v = 2t$ --> $t = \frac{v}{2}$

so, $x=\frac{v^2}{4}$

Method 2:

$a = x''(t) = 2$

$v*\frac{dv}{dx} = 2$

$\frac{dv}{dx} = \frac{2}{v}$

$\frac{dx}{dv} = \frac{v}{2}$

$x = v^2/4 + C$

These differ by a constant, why? Where exactly is the error in method 1? I presume that by substituting, you are rejecting additional information. BUT, I don't see where the error has been made.

Answer 1

The reason you get different answers is because you differentiated each side again. When you differentiate two equations you lose information in general. In this case, you lose information about the constant term.

Both results are the same equation, but in the second case you need to find the value of $C$ that makes the equation true using information you already have. You could, for example, plug in $x=t^2$ and $v=2t$ to solve that $C=0$, but this is basically what you already did in the first problem. This shows that there isn't much worth to solving it in your second method.