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 | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {y^2} 2\) | \(=\) | \(\ds k x + \frac C 2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds y^2\) | \(=\) | \(\ds 2 k x + C\) |
$\blacksquare$