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 \( \ddot x + (\frac{dx}{dt})^3 = x \tan t \).

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

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