First Order ODE/y dy = k dx

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Theorem

Let $k \in \R$ be a real number.

The first order ODE:

$y \, \dfrac {\d y} {\d x} = k$

has the general solution:

$y^2 = 2 k x + C$


Proof

\(\ds y \, \dfrac {\d y} {\d x}\) \(=\) \(\ds k\)
\(\ds \leadsto \ \ \) \(\ds \int y \rd y\) \(=\) \(\ds \int k \rd x\) Solution to Separable Differential Equation
\(\ds \leadsto \ \ \) \(\ds \frac {y^2} 2\) \(=\) \(\ds k x + \frac C 2\)
\(\ds \leadsto \ \ \) \(\ds y^2\) \(=\) \(\ds 2 k x + C\)

$\blacksquare$