Problems On Application Of Partial Differential Equation

Problems on Application of Partial Differential Equation-1:

Example 2. Classify, reduce to normal form and obtain the general solution of the partial differential equations

For this equations The equation is parabolic everywhere in the plane (x, y). The characteristic equations is

Therefore there is one family of characteristic y/x = constant.

Let ξ = x and η = y/x. Then using the chain rule.

Substituting in the partial differential equations we obtain the normal form uξξ = 4

Integrating with respect to ξ

Where f is an arbitrary functions of a real variables. Integrating again with respect to ξ

Therefore the general solution is