When doing 49, we can see that we have AY and AY prime and AY double prime. So we're going to let our Y prime be P and remember that Y prime is really DYDX. So then our Y double prime is DPDX. But because this was all in terms of Y and we had just plain YS, we have to think about using the chain rule. So DPD y * d YDX that DYDX was really just P. So this turns into P * d PDY. So now if we just do our straight substitution, we're going to get YPDPDY plus P ^2 equaling YP. I'm going to divide everything through by YP so we get DPD y + P / y equaling 1. So then we can see that our row is going to be E to the integral of 1 / y DY or E to the LNY or just plain Y So we're going to multiply everything 3 by that Y. So we get YDPDY plus just plain P equaling Y. So when we think about the derivative of this, this is just y * P equaling Y. Now from here we're going to take the integral in terms of DY, and if we do one side, we have to do the other. So this is going to turn into YP equaling one half y ^2 plus some constant. If we divide everything through by Y, we get Y or P equal 1 half y + C / y We thought about getting that as a common denominator. We'd get y ^2 plus 2C over 2Y. We could call that two CA new variable, say 2C1. So now that P is really really just RDYDX. So we get 2Y DX equaling y ^2 plus some C 1 / 2 Y. So then two y / y ^2 plus C one times the DY is equaling the DX. And if we integrate both sides we can see by just a straight U sub that the derivative of the bottom IS2Y which is the top. So this side is lane of y ^2 plus our C1, and this side is just plain old X to get rid of actually X plus some C2 to get rid of the natural log. We're going to have y ^2 + C one equaling E to the X + C two. We can think of this as E to the XE to the C2. And then we can say this E to the C2 is some new constant, and it could be positive or negative. So what we're going to be able to do then is to drop the absolute value and we're going to get y ^2 + C one is going to equal some a * e to the X. So then we could just solve for this and say y ^2 equal AE to the X -, C one.