Existence-Uniqueness Theorem for Homogeneous First-Order Differential Equation

Theorem
Let $\map P x$ be a continuous function on an open interval $I \subseteq \R$.

Let $a \in I$.

Let $b \in \R$.

Let $\map f x = y$ be a function satisfying the differential equation:
 * $y' + \map P x y = 0$

and the initial condition:
 * $\map f a = b$

Then there exists a unique function satisfying these initial conditions on the interval $I$.

That function takes the form:
 * $\map f x = b e^{-\map A x}$

where:
 * $\ds \map A x = -\int_a^x \map P t \rd t$

Existence
Differentiating $\map f x = b e^{-\map A x}$ $x$:

So thedifferential equation becomes:
 * $\map {f'} x + \map P x \map f x = -\map P x \map f x + \map P x \map f x = 0$

For the initial condition:

Thus such a function exists satisfying the conditions.

Uniqueness
Suppose that $f$ is a function satisfying the initial conditions.

Let $\map g x = \map f x e^{\map A x}$.

By Product Rule for Derivatives:

So $\map g x$ must be constant.

Therefore:
 * $\map g x = \map g a = \map f a e^{\map A a} = \map f a = b$

From this, we conclude that:
 * $\map f x = \map g x e^{-\map A x} = b e^{-\map A x}$