8.1.1 Intro to Differential Equations

What Are Differential Equations?

A differential equation (D.E.) is an equation that relates a function and its derivatives. They appear in a wide range of applications, from physics to economics, describing how quantities change over time or space.

Ordinary Differential Equations (ODEs

Involve derivatives of a function with respect to a single variable (e.g., $x$)

Partial Differential Equations (PDEs)

Involve partial derivatives of a function with respect to multiple variables (e.g., $x$ and $y$).

Types of First-Order Differential Equations

Separable Equations:

Can be written in the form:

$$\frac{dy}{dx} = g(x)h(y)$$

These are solved by separating the variables and integrating:

$$\int \frac{1}{h(y)} \, dy = \int g(x) \, dx$$

Linear Equations:

Have the form:

$$\frac{dy}{dx} + P(x)y = Q(x)$$

These can be solved using an integrating factor $F(x)$, where:

$$F(x) = e^{\int P(x) \, dx}$$

Exact Equations:

Have the form:

$$M(x, y) \, dx + N(x, y) \, dy = 0$$

where $M$ and $N$ satisfy:

$$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$$