Question

Q2. Consider the equation (a) [2 marks] Find the characteristics of the equation. (b) [4 Marks] Sketch the characteristics in the (x,y) plane (c) [2 Marks] determine the characteristic coordinates (d) [6 marks] Reduce the equation to standard form and find its general solution (e) Use the general solution to find u(x, y), if it exists, for the following Cauchy data () [2 Marks] u(x,y)-2 on the curve y=x2 [2 Marks] u(x,y)-l on the curve y- (c) [2 Marks) u(x,y)-1 on the curve y-x

Answer 1

SOLUTION:

NOW WE DIVIDE THE GIVEN EQUATION WITH X THEN WE GET THE EQUATIONS AS

NOW SOLVATION GOES AS:

Consider the partial differential equation Lagrange's auxiliary equation is as follows dx dy dr dy r)

du dr = r(1-u) du (1-u) 12 dr = (u-1)

C2 u-1 ез The general solution is as follows ез