Question

Find the most general function M(x,y) for the equation:
M(x,y)dx +(x sec^2y- x/y)dy=0
so that the equation is exact

Answer 1

is
M(x,y) =tany -lny

Answer 2

For the equation 2 b exact, dM/dy must be equal to dN/dx.
So, my 1st try is find dN/dx where\[N = ( x \sec ^{2}y -\frac{ x }{ y})\]
therefore \[\frac{ dN }{ dx }= \sec ^{2}y -\frac{ 1 }{ y }\]

Answer 3

yup

Answer 4

Hw to get that ans? Can u pls explain further?

Answer 5

just integrate both sides

Answer 6

So the next step is integrating dN/dx?

Answer 7

yup

Answer 8

and hence N=tany -lny +c

Answer 9

but it should be dN/dy