SKEDSOFT

Problems on Application of Partial Differential Equation:

Example 1. Classify the following linear second order partial differential equation and find its general solutions:

In this example the partial differential equations is hyperbolic provided x ≠ 0 and parabolic for x = 0. For x ≠ 0 the characteristic equations are:

If y' = 0, y = constant. If y' = x/y, x² - y² constant. Therefore two families of characteristic are

Using the chain rule a number of times we calculate the partial derivatives.

Substitute into the partial differential equations we obtain the normal form.

Where f is an arbitrary function of one real variables. Integrating again with respect to ξ

Where f and g are arbitrary functions of one real variables. Reverting to the original coordinates we find the general solutions.