First Order ODE/y' + y = 0

Theorem

The first order ODE:

$\dfrac {\d y} {\d x} + y = 0$

has the general solution:

$y = C e^{-x}$

where $C$ is an arbitrary constant.

Proof

This first order ODE is in the form:

$\dfrac {\d y} {\d x} + k y = 0$

where $k = 1$.

From First Order ODE: $\d y = k y \rd x$, this has the solution:

$y = C e^{-x}$

Sources

• 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $3$: The Differential Equation: $(3.11)$