Question

Differential equations

2.4.28 For each of the following equations, find the most general function N(xy) so that the equation is exact (a) Select the correct choie below and fill in the answer box to complete your choice. Type an expression using x and y as the variables) O A.ho), where ho) is an arbitrary function of x OB, ] +g(y), where gy) is an arbitrary function ofy.

Answer 1

[3y cos (xy) + e^x] dx + N(x , y) dy = 0 ..(1) comparing eqn (1) by M dx + N dy we get M = 3y cos (xy) + e^x & N = N (x , y) for exactness partial differential M/ partial differential y = partial differential N/ partial differential x so partial differential M/ partial differential y = 3(y (- sin (xy) . x) + cos xy) partial differential M/ partial differential y = partial differential N/partial differential x Rightarrow partial differential N/partial differential x = 3 [cos xy - 3 xy sin xy] Rightarrow N = integral 3 [cos xy - xy sin xy] dx + phi (y) = 3 [sin xy/y - y (- x cos xy/y + sin xy/Y^2)] + phi (y) = 3 [sin xy/y + x cos xy - sin xy/y] + phi (y) N = 3x cos xy + phi (y)