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
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