# First Order ODE/x y' = 2 y

## 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 \ \$ $\ds y$ $=$ $\ds C x^2$

$\blacksquare$