First Order ODE/x y' = 2 y

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Theorem

The first order ODE:

$x y' = 2 y$

has the general solution:

$y = C x^2$

where $C$ is an arbitrary constant.


Proof

\(\ds x y'\) \(=\) \(\ds 2 y\)
\(\ds \leadsto \ \ \) \(\ds \int \dfrac {\d y} y\) \(=\) \(\ds 2 \int \dfrac {\d x} x\) Separation of Variables
\(\ds \leadsto \ \ \) \(\ds \ln y\) \(=\) \(\ds 2 \ln x + \ln C\) Primitive of Reciprocal
\(\ds \leadsto \ \ \) \(\ds \ln y\) \(=\) \(\ds \map \ln C x^2\)
\(\ds \leadsto \ \ \) \(\ds y\) \(=\) \(\ds C x^2\)

$\blacksquare$


Sources