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## Homework Statement

Determine whether exact. If yes, solve.

(4t

^{3}y-15t

^{2}-y)dt + (t

^{4}+3y

^{2}-t)dy = 0

## The Attempt at a Solution

M= (4t

^{3}y-15t

^{2}-y)

N= (t

^{4}+3y

^{2}-t)

M

_{y}=4t

^{3}-1

N

_{t}= 4t

^{3}-1

So, yes it is exact.

f

_{y}= M = (4t

^{3}y-15t

^{2}-y)

f(t,y) = ∫ N dy

= 2t

^{3}y

^{2}-15t

^{2}y-(1/2)y

^{2}+g(t)

f

_{t}= 6t

^{2}y

^{2}-30ty+ g'(t)

This is where I've gotten stuck. I know I need to set this equal to M, and then all of the t's should cancel, but from what I've done, that won't work. So, this means I've made a mistake somewhere else.

Thank you!