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Solve the differential equation.$$2 \sqrt{x y} \frac{d y}{d x}=1, \quad x, y>0$$

$$\frac{2}{3} y^{3 / 2}-x^{1 / 2}=C$$

Calculus 1 / AB

Calculus 2 / BC

Chapter 7

Integrals and Transcendental Functions

Section 2

Exponential Change and Separable Differential Equations

Functions

Trig Integrals

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

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we want to solve this differential equation here, so this will end up being a separable equation, and it may help us see that if we were to divide each side by the square root of necks so that's going to give us two square to buy d Y is equal to, and I'm going to rewrite this as X to the negative 1/2. I should still be over DX. But now let's go ahead and multiply d X to the other side. I say multiply dx in quotes because we really can't multiply. But it works out for other reasons. So, uh, DX on this side now we could go ahead and integrate each side and let me rewrite Screwed. Why? As why to the 1/2 power. So now we could go ahead and use powerful to integrate excite. So on the left, Where to? Why now this to the three house power that we want to multiply by 2/3 and this is going to be equal to so x two, the 1/2 power now and then we would multiply by. Two guys were multiplying by are really dividing by the new power and then we have our constant C. Right. So now let's do a little bit of manipulations, toe, get some numbers to cancel out. So noticed that we can first let me expand screen and spewed all this over. Actually, um, it's, um multiply we get or thirds why? 2 3/2 secret to X. I'm gonna rewrite that as square root of ex. Now, now, with this point, we can multiply each side by re force and move. It appears that's gonna give us why is too re haps or why 23 house and then multiplying to buy 34 should give us three halfs square root of X and then we're gonna have C times three halves. So that's just gonna be another constant. So I'll call this 1st 1 C one. So this one here will be see too. And now we would want to undo our power here. And we would do that by raising each side to the 2/3 power which is gonna give why is equal to 3/2 square root of X waas. See, too, All to the two herbs power. And so now, since we solved for why, let's just go ahead and get rid of that C two and call that our new C. So this will be our solution to this differential equation.

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