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# Particular solutions to differential equations: rational function

AP.CALC:
FUN‑7 (EU)
,
FUN‑7.E (LO)
,
FUN‑7.E.1 (EK)
,
FUN‑7.E.2 (EK)
,
FUN‑7.E.3 (EK)

## Video transcript

so we're told that F of 2 is equal to 12 f prime of X is equal to 24 over X to the third and what we want to figure out is what is F of negative 1 alright so they give us the derivative in terms of X so maybe we can take the antiderivative of the derivative to find our original function so let's do that so we could say that f of X f of X is going to be equal to the antiderivative or we could say the indefinite integral of F prime of X which is equal to 24 over X to the third I could write it over like this 24 over X to the third but to help me process it a little bit more I'm going to write it I'm going to write this is 24 X to the negative 3 because then it'll become a little how to take that anti derivative d/dx and so what is the antiderivative of 24 X to the negative 3 well we're just going to do the power rule in Reverse so what we're going to do is we're going to increase the exponent so let me just rewrite it it's going to be 24x to the we're going to increase the exponent by 1 so it's going to be X to the negative 3 plus 1 and then we're going to divide by that increased exponent so negative 3 plus 1 and so that is going to be negative 3 plus 1 is X to the negative 2 and then we divide by negative 2 and if you're in doubt about what we just did where we're kind of doing the power rule in Reverse now take the power rule take the derivative of this using the power rule negative 2 times 24 over negative 2 it's just going to be 24 and then you decrement that exponent you go to negative 3 so are we done here is this f of X well f of X might involve a constant so let's put a constant out out here because notice if you were to take the derivative of this thing here the derivative of 24 X to the negative 2 over negative 2 we already established is 24 X to the negative 3 but then if you take the derivative of a constant well that just disappears so you don't see it when you look at the derivative so we have to make sure that there might be a and I have a feeling based on the information that they've given us we're going to make use of that constant so let me rewrite f of X so we know that f of X can be expressed as 24 divided by negative 2 it's negative 12 X to the negative 2 plus some constant so how do we figure out that constant well they have told us what F of 2 is f of 2 is equal to 12 so let's write this down so when so f so f of 2 is equal to 12 which is equal to well we just have to put 2 in everywhere we see an X that's going to be negative 2 times 2 to the negative 2 power plus C and so 12 is equal to what is this 2 to the negative 2 2 to the negative 2 is equal to 1 over 2 squared which is equal to 1/4 so this is negative 12 times 1/4 negative 12 times 1/4 is negative 3 so it's negative 3 plus C now we can add 3 to both sides to solve for C we get 15 is equal to 15 is equal to C so or C is equal to 15 that is equal to 15 and so now we can write our f of X as we get f of X is equal to negative 12 and I could even write that as negative 12 over x squared if we like negative 12 over x squared plus 15 and now using that we can evaluate F of negative 1 F of negative 1 wherever we see an X we put a negative 1 there so this is going to be negative 1 squared so F of negative 1 is equal to 12 divided by negative 12 divided by negative 1 squared well negative 1 squared is just 1 so it's going to be negative 12 plus 15 which is equal to 3 and we're done this thing is equal to 3