Question

What is the characteristic equation of pq=xy and determine the integral surface which passes through curves z=x,y=0?

a) p+q=0, z=x^2
b) p−q=0, z=x^2
c) p+q=0, z=x
d) p−q=0, z=x

Answer 1

The characteristic equation of the given PDE pq = xy is p = x, q = y. The integral surface is z = x^2 that passes through the curve z = x, y = 0, making (b) p - q = 0, z = x^2 the correct answer.

Step-by-step explanation:

The given partial differential equation (PDE) is PQ = XY, which implies that the characteristic equation is obtained by setting each of the products equal to each other, i.e., p = x and q = y or vice versa. To satisfy the integral surface condition through the given curve z = x and y = 0, we use the method of characteristics. We first note that along the curve, the variables can be parameterized such that z (0) = x (0), and hence we are looking for a solution that extends from this initial condition.

When we integrate p = DZ/DY and q = DZ/DY with the initial condition z = x along the curve y = 0, we deduce that the integral surface compatible with this is z = x2, which implies that the derivative DY/DZ= 2x = p, and since q = y and y = 0 along the initial curve, we have p - q = 0.

Therefore, the correct answer is (b) p - q = 0, z = x2.