Question

Use the method of characteristics to find solutions to the following first-order partial differential equa- tions (PDEs): V_zī ar +Ou-0, (a) [10 points] u(0,y)-y. (b) [10 pointsj (0,y) - sin y

Answer 1

Given squareroot 1 - x^2 partial differential u/partial differential x + partial differential u /partial differential y = 0 - (*) with u(0, y) = y so characteristics equation are dx/ds = squareroot 1 - x^2: dy/ds = 1, dz/ds = 0 where s is parameter integrating eliminating s from (1) and (2) = > sin^- 1 x = s + c_1 - (1), y = s + c_2 - (2) , z = c_3 (1) and (2) = > sin^- 1 x - y = c_4 and z = c_3 = > z(x, y) = f(sin^- 1 x - y) u(x, y) = f(sin^- 1 x - y) for any smoothly summation of clearly it satisfies (*) since squareroot 1 - x^2 times f'(sin^- 1 x - y) = 1/ squareroot 1 - x^2 + f'(sin^- 1 x - y) given condition u(0, y) = y (- 1) = 0 = > f(- y) = y let phi (- y) = y let (Gamma, phi) be the curve in R^3 given by {(0, y, phi(- y)} = > x(0) = 0, y(0) = y_0, z(0) = phi(- y_0) let particularization of Gamma by s then Gamma = {(0, s)}x (0, s) = 0 y (0, s) = 0