Answer Happy

Consider the ODE written in differential form {126 cos(7x) sin(3y)} dx + {54 sin(7x) cos(3y)} dy = 0 (a) The expression for the test of exactness is My = Nx = O 126 cos(7x) sin(3y) O 378 cos(7x) cos(3 y) 18 sin(7x) sin(3y) O 54 sin(7x) cos(3y) The implicit solution to the ODE, u(x, y) = C, can be found by solving a system of two PDEs. (b) What is the expression for ux? 378 cos(7x) cos(3 y) 126 cos(7x) sin(3y) O 18 sin(7x) sin(3 y) O 54 sin(72) cos(3y) (c) What is the expression for uy? O 378 cos(7x) cos(3 y) O 18 sin(7x) sin(3 y) O 54 sin(7x) cos(3y) O 126 cos(7x) sin(3 y)

— с (d) The implicit solution is 126 cos(73) sin(3y) 378 cos(7x) cos(3 y) = 54 sin(7 x) cos(3 y) = C 18 sin(7x) sin(3 y) = C =C = = where c is an arbitrary constant. The following hint maybe helpful in some instances (i.e. your individually randomised problem): If u(x, y) = c then u(x, y) + k=c is also a solution, where k is a constant.