If **y** is a function of** x,** and **no** is a positive integer, so an equality relationship (not reducible to an identity) involving x, y, y ', y ",…, y^{(n)} it's called a **differential equation of order n.**

Differential equation is an equation that presents derivatives or differentials of an unknown function (the unknown of the equation). |

## Classification

- Ordinary Differential Equation (ODE): Involves derivatives of a single independent variable function.
- Partial Differential Equation (EDP): Involves partial derivatives of a function of more than one independent variable.

**Order**: is the order of the highest order derivative of the unknown function that appears in the equation.

### Examples

y '= 2x | have order 1 and grade 1 |

y "+ x^{2}(y ')^{3} - 40y = 0 | have order 2 and grade 3 |

y "'+ x | have order 3 and grade 3 |

### Resolution

The solution of a differential equation is a function that contains neither derivatives nor differentials and satisfies the given equation (ie, the function that, substituted in the given equation, transforms it into an identity).

Ex: Ordinary Differential Equation: = 3x^{2} - 4x + 1

dy = (3x^{2} - 4x + 1) dx

dy = 3 x^{2}dx - 4 xdx + dx + C

y = x^{3} - 2x^{2} + x + C (**general solution**)

An **particular solution **can be obtained from the general through, for example, the condition y (-1) = 3

(initial condition)

3 = -1 - 2 - 1 + C C = 7 y = x^{3} - 2x^{2} + x + 7 (private solution)

Note: In either case, the proof can be done by deriving the solution and thereby returning to the given equation.

The solutions fall into:

* General solution* - presents n constants independent of each other (n = ODE order). These constants, as appropriate, may be written C, 2C, C

^{2}lnC

** Particular solution** - Obtained from the general under given conditions (called initial conditions or boundary conditions).