First Order ODE/y' = e^x

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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

$\blacksquare$


Sources