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

Show Answer
The order is the highest derivative which is 2 on the term \(\ddot x\). The degree is the largest exponent which is on 3 on term \((\frac{dx}{dt})^3\).

A differential equation is **linear** if it is linear in terms of \( y, y^{'}, \dots y^{(n)} \). These terms can have coefficients
with non-linear terms of the __independent variable__. The general form of a linear DE is:

\( a_n(x) \cfrac{d^ny}{dx^n} + a_{n-1}(x) \cfrac{d^{n-1}y}{dx^{n-1}} + \dots + a_1(x) \cfrac{dy}{dx} + a_0(x) y = f(x) \)

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

Show Answer
The equation is **linear**. Although there is an \(x^2\), it is non-linear w.r.t to independent variable so it doesn't impact the linearity.
Same goes for \(e^{2x} \).