# First Order ODE/y' = e^x

## Theorem

The first order ODE:

$y' = e^x$

has the general solution:

$y = e^x + C$

where $C$ is an arbitrary constant.

## Proof

 $\ds y'$ $=$ $\ds e^x$ $\ds \leadsto \ \$ $\ds \int \d y$ $=$ $\ds \int e^x \rd x$ Separation of Variables $\ds \leadsto \ \$ $\ds y$ $=$ $\ds e^x + C$ Primitive of Constant, Primitive of Exponential Function

$\blacksquare$