Partial Differential Equations

Introduced in Semester 1.

Notations

When the case of 2 independent variables is considered, and z$ is assumed be the dependent variable. The following notations are used in this section.

If there are independent variables, they are considered as . In this case the following notations are used:

Types

Suppose .

Linear

When it is linear in . When it is of form:

Semi-linear

When it is linear in and are functions of only. When it is of form:

Quasi-linear

When it is linear in and . Whe it is of form:

Non-linear

When it does not satisfy any of the above type.

Lagrange’s Equation

Quasi-linear partial differential equation of order one.

Solving method

General solution of Lagrange’s equation, is an arbitrary function. And and are two independent solutions of:

Here and are arbitrary constants. At least one of or must contain . and are independent if is not a constant.