# Introduction to Differential Equations

A Differential Equation is an equation that involves derivatives of functions with respect to independent variables.

Examples:

• $$\frac{dy}{dx}=-y+x$$
• $$\frac{d^2y}{dx^2}-y=\sin(\pi x)$$
• $$(\frac{du}{dx})^2=-(\frac{du}{dy})^2$$

## Notation

Derivatives can be written using different notations summarized below:

 Notation Examples Leibniz $$\cfrac{dy}{dx}, \cfrac{d^3u}{dv^3}$$ Prime $$y^{'}, u^{''}, v^{(4)}$$ Dot $$\dot{x}, \ddot{y}$$ Subscript $$\cfrac{\partial^2y}{\partial x \partial t }= \cfrac{\partial}{\partial x} (\cfrac{\partial y}{ \partial t}) = y_{tx}$$

## Types of Differential Equations

 Ordinary Differential Equations (ODEs) contain only ordinary derivaties with respect to a single independent variable. $$\cfrac{dy}{dx} + 3y - \cos t = 0$$ $$\ddot x - 2\dot x = e^t$$ Partial Differential Equations (PDEs) contain partial derivaties with respect to several independent variable. $$\frac{du}{dx} - 3 \frac{du}{dy} = 0$$ $$v_x \sin t + v_y \cos t = v_{tt}$$

The order of a differential equation is the order of the highest derivative in the equation. The degree of a differential equation is the degree of the highest term.

Determine the order and degree of $$\ddot x + (\frac{dx}{dt})^3 = x \tan t$$.

Determine whether the DE is linear $$2xy^{'''} -5x^2y^{'} = e^{2x}$$.