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We will classify these equations into three different categories. If b2 − 4ac > 0, we say the equation is hyperbolic. If b2 − 4ac = 0, we say the equation is parabolic. If b2 − 4ac < 0, we say the equation is elliptic.

Systems of equations can be classified by the number of solutions. If a system has at least one solution, it is said to be consistent . If a consistent system has exactly one solution, it is independent . If a consistent system has an infinite number of solutions, it is dependent .

The order of a differential equation is defined to be that of the highest order derivative it contains. The degree of a differential equation is defined as the power to which the highest order derivative is raised. The equation (f‴)2 + (f″)4 + f = x is an example of a second-degree, third-order differential equation.

The different types of differential equations are: Ordinary Differential Equations. Homogeneous Differential Equations. Non-homogeneous Differential Equations. Linear Differential Equations. Nonlinear Differential Equations.

A system of two equations can be classified as follows: If the slopes are the same but the y-intercepts are different, the system is inconsistent. If the slopes are different, the system is consistent and independent. If the slopes are the same and the y-intercepts are the same, the system is consistent and dependent.

Partial differential equations occur in many different areas of physics, chemistry and engineering. Second order P.D.E. are usually divided into three types: elliptical, hyperbolic, and parabolic.

We can place all differential equation into two types: ordinary differential equation and partial differential equations. A partial differential equation is a differential equation that involves partial derivatives. An ordinary differential equation is a differential equation that does not involve partial derivatives.

Classification of Differential Equations Partial Differential Equations. Linear Differential Equations. Non-linear Differential Equations. Homogeneous Differential Equations.