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

\(\ds y'\)

\(=\)

\(\ds e^x\)

\(\ds \leadsto \ \ \)

\(\ds \int \d y\)

\(=\)

\(\ds \int e^x \rd x\)

Solution to Separable Differential Equation

\(\ds \leadsto \ \ \)

\(\ds y\)

\(=\)

\(\ds e^x + C\)

Primitive of Constant, Primitive of Exponential Function

$\blacksquare$

Sources

• 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $3$: The Differential Equation: $(3.12)$