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let's say we need to find a solution to the differential equation that the derivative of Y with respect to X is equal to x squared over e to the Y pause this video and see if you can have a go at it and I will give you a clue it is a separable differential equation alright now let's do this together so whenever you do any differential equation the first thing you should try to see is is it separable and when I say separable I mean I can get all the expressions that deal with Y on the same side as the dy and I can separate those from the expressions that deal with X and they need to be on the same side as my differential DX so how can we do that well if we multiply both sides times e to the Y then e to the Y will go away right over here so we will get rid of this Y expression from the right hand side and then we can multiply both sides by DX so if we did that so let me move my screen over a bit to the left so I'm going to multiply both sides by e to the Y and I'm also going to multiply both sides by DX times DX and multiplying by DX gets rid of the DX on the left-hand side and it sits on the right-hand side with the x squared and so all of this is now e to the Y dy is equal to x squared x squared DX and just the fact that we were able to do this shows that is acceptable now what we can do now is integrate both sides of this equation so let's do that so what is the integral of e to the Y dy well one of the amazing things about the expression or you could say the function but something is equal to e to the ax normally we say e to the X but in this case it's e to the Y is that the antiderivative of this it's just e to the Y we've learned that in multiple videos I always express my fashion fascination with it so this is just e to the Y and likewise if you took the derivative of e to the Y with respect to Y it would be e to the Y remember this were because we are integrating with respect to Y here so the integral of e to the Y with respect to Y well that's e to the Y and so that is going to be equal to the antiderivative of x squared well that is we increment the exponent so that gets to X to the third power and we divide by that incremented exponent and since I took the indefinite integral of both sides I have to put a constant on on at least one of these sides so let me throw it over here so plus C and just to finish up especially it's a lot of examinations like the AP exam they might want you to write in the form where Y is explicitly is an explicit function of X so to do that we can take the natural log of both sides so we take the natural log of that side and we take the natural log of that side well the natural log of e to the Y what power do we have to raise e to to get to e to the Y well that's why we took the natural log this just simplifies as Y and we get Y is equal to is equal to the natural log of what we have right over here X to the third over 3 plus C and we are done