Question

Consider the PDE 2u, - 3uy = 0 where u = u(x,y). It can be shown (you don't have to do it) that product solutions to the PDE take the form u(x, y) = X(1)Y(y) = Cetrefty, (1) where C and can be any pair of constants. Here is what you need to work on: First, find infinitely many solutions to the PDE that look different from 1. Make sure to mention which method you used, if any, and explain how you got solutions. You might check materials beyond lecture notes and reference textbooks, but you could solve these questions without doing so. Second, what's the difference, in terms of answers, between solutions of the form (1) and solutions you obtained?

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Answer 1

0 UCXy) = X(X) YLY) = Ceanta gatxe Answer: *Consider from the given information: e @uz-344 : 02 Where, cand can be any pair of Constants ble can use method of characterstics for partial diferential squahon. to sdve the given a Equation: - Qdy +3dx=0 The et en die dae is the characterstic Equation or Consider the first twofinchions of 3 days ago integrate the above equation then we get, ay+323C, {where, cis arbitary Constantį Consider the last two frachions of we get al. dy > duzoo u=C2 ge hoval solution of give patial differentias Gzs ckbrary Constant Equation in ☺ is uCX, Vf(34+24) {where, f is a orbitan differentiable * By observing equahon o the solution & Exporentiae feinction. function in one variable? If we go with the enethod of charactershire we can find a more genover general Solution Thus, in the mothod of characterstic we obtain a solution in terms of a orbitary differential funchou.