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Differential equation

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Slope Field

Possible solutions to the differential equation \frac{dy}{dx}=x^2-x-2 represented by a slope field.

A differential equation is an equation which relates a function to at least one of its derivatives. If the function in question has only one independent variable, the equation is known as an ordinary differential equation; if the function is of multiple variables, it is called a partial differential equation. Generally, differential equation

The basic concept of a differential equation is that it relates the trends in the function to other trends in the function. That is to say, a differential equation relates derivatives of varying degrees (slopes, gradients) to one another and to other functions of x.

The study of differential equations is an extension of differential and integral calculus.

Objective

When given a differential equation, the objective is often to find the function y(x) as an explicit function in terms of only x, and possible functions of x; sometimes only implicit functions of y(x) are possible; but necessarily the objective is to remove all derivatives of y(x) from the equation.

For example, the differential equation:

y'' - 5y' + 6y = 0 \;\;\;\; y(0) = 3; \;\; y'(0) = 7

Will have the following solution:

y(x) = 2e^{2x} + e^{3x}

Types of differential equations

As previously stated, differential equations are divided into ordinary (ODE) and partial differential equations (PDE).

The degree of the equation is the exponent of the highest order derivative, and the order is the order of the highest order derivative. If the function and its derivatives are all to the power of one (making the equation first degree), and the function is not multiplied with any of its derivatives, the equation is considered linear. Nonlinear differential equations are much more difficult to solve, if it is possible to solve them directly at all.

Differential equations are homogeneous if they meet the condition that if f(x) is a solution, so is cf(x), where c is an arbitrary constant. This condition can only be met if every term has the dependent variable or a derivative of it.

The solutions to linear, homogeneous differential equations form a vector space.

Applications

Navier Stokes Laminar

The Navier–Stokes differential equation, used to model airflow around an obstruction.

Differential equations are highly applicable to the real world. Often times, be it in statistics, economics or theoretical physics, data will correlate in such ways that determining a differential equation is more easily done than trying to extrapolate a direct function.

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