Classification of PDEs is an important concept because the general theory and methods of solution usually apply only to a given class of equations. Let us first discuss the classification of PDEs involving two independent variables.

Consider the following general second order linear PDE in two independent variables:

where and are functions of the independent variables and . This equation may be written in the form.

,

where

Assume that and are continuous functions of and possessing continuous partial derivatives of as high order as necessary.

The classification of PDE is motivated by the classification of second order algebraic equations in two-variables

We know that the nature of the curves will be decided by the principal part i.e., the term containing highest degree. Depending on the sign of the discrimination , we classify the curve as follows:

- If then the curve traces
**hyperbola**. - If then the curve traces
**parabola**. - If then the curve traces
**ellipse**.

With suitable transformation, we can transform into the following normal form:

- For –> hyperbola
- For –> parabola
- For –> ellipse