Differential Equations

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).


  • 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.


y '= 2x

have order 1 and grade 1
y "+ x2(y ')3 - 40y = 0 have order 2 and grade 3

y "'+ x2y3 = x.tanx

have order 3 and grade 3


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: = 3x2 - 4x + 1

dy = (3x2 - 4x + 1) dx

dy = 3 x2dx - 4 xdx + dx + C

y = x3 - 2x2 + 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 = x3 - 2x2 + 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, C2lnC

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

Next: Homogeneous Linear Equations, 2nd Order