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Solve the differential equation.$$\frac{d y}{d x}=\sqrt{y} \cos ^{2} \sqrt{y}$$

$$-x+2 \tan \sqrt{y}=C$$

Calculus 1 / AB

Calculus 2 / BC

Chapter 7

Integrals and Transcendental Functions

Section 2

Exponential Change and Separable Differential Equations

Functions

Trig Integrals

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Solve the differential equ…

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we want to solve this differential equation here. So let's go ahead and separate this because if we divide bye square root of why and co sign square squared of what We should end up with one over the square root of why and now one of her co Sinus seeking. So if we have that squared, this should be seeking squared square root of why and we'll have d y on the left side. Now if we were to multiply over bye dx, we would have won times DX now we would just integrate each side so the right hand side would integrate two x plus are constant And now to integrate the left side over here, we're gonna need to do a substitution that looks like So let's let you eating two square. Why taking the drift of that? Because this d'you is equal to 1/2 times one over the square root of I Just by using power, no nose if we multiply each side of this by two, the 1/2 counsels on the right and what we have here is exactly that right there. So that means we can rewrite this interval as C can't squared you times too deep. And we know that the integral of seeking is Cosa. Um is tension I was thinking sign for some reason. Uh, yes. So that should be tended You. So we'll have to Tanja you. But we don't care about you. We care about square root of why So we plug that in and this is even two x plus c. So this here would be our social so we could go and solve for why, But once we take like, Arc Tangent is gonna get really messy, so I think this would be the best place to leave it.

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Harvey Mudd College

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