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In Exercises $1-10,$ find the general solution to the exact differential equation.$$\frac{d y}{d u}=4(\sin u)^{3}(\cos u)$$

$$y=\sin ^{4} u+c$$

Calculus 1 / AB

Calculus 2 / BC

Calculus 3

Chapter 6

Differential Equations and Mathematical Modeling

Section 1

Slope Fields and Euler's Method

Differentiation

Integration Techniques

Differential Equations

Second-Order Differential Equations

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So this problem wants us to find the general solution to the exact differential equation. And this problem is kind of hunting at something we learned with future, tempting to use substitution. Um, but what we really have to do is if we integrate, do you? Why do we get why equals? And then we're looking at this for sign, You Cube Times co sign You should remind us of the chain roll because the derivative of sign You is co sign you so we can think of it as what if we had? Why equal this Something that when we took the derivative of we had to use the chain rule too. Ah, multiplied by the derivative. What was are you so we could think of it as our d y Do you equalled for you cubed And then Arcos I knew is are do you so if we integrate, we get you to the fourth. So if are you were thinking of us, sign you And I'm just using a little confusing. But we could think of it as like ze right here we have four z cubed easy to use the four Just go do some random variable we have zero The four where z assign you weaken Say that y equals sign you to the fourth And we could check by taking the derivative between the derivative deride you We take that for being down afore sign you cubed And then we have to use the chain rule to worldwide by the derivative of this getting inside which is close on you. And there we go. We have or do I D. X. But again, we can't forget, as always when buying a general solution that there's a possibility the constant. So our answer is why equals signed in the fourth plus C.

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