Question

We are given a differential equation and will rewrite it in the form M(x, y) dx + N(x, y) dy = 0. xy2 dy dx = y3 − x3 xy2 dy = (y3 − x3) dx xy2 dy − (y3 − x3) dx = 0 xy2 dy + (−y3 + x3) dx = 0

Answer 1

Check attachment for orderliness of question

Explanation:

We want to the equation in an exact form

M(x, y) dx + N(x, y) dy = 0.

xy² dy/dx = y³− x³

Cross multiply

Then,

xy²dy=(y³-x³)dx

xy²dy - (y³-x³)dx = 0

xy²dy + (x³-y³)dx=0

Therefore,

(x³-y³)dx + xy²dy=0

Can only be an exact equation if and only if

dM/dy=dN/dx

From the equation

M=(x³-y³)

Then, dM/dy=-3y²

Also, N=xy²

dN/dx=y²

Since dM/dy ≠ dN/dx

Then it is not an examct equation and we can say

M≠(x³-y³)

And

N≠(xy²)