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Find the function $ f $ such that $ f'(x) = xf(x) - x $ and $ f(0) = 2. $

$f(x)=e^{x^{2} / 2}+1$

Differential Equations

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this question asked us to find the function f Such a f prime of X is X f of X minus acts and then given the fact that F zero is too now, first things first. We know we can right f prime of acts in terms of de roi de axe. It's easier to use why instead of F because then we can later on integrate our variables factor the right hand side to make it easier to divide. Now we can transfer all the Y terms over to the left hand side and all the ex terms over to the right hand side. Take the integral of both sides Integrate. Remember, we increased the extranet by one as you can see it from X to X squared and then we divide by the new exponents which is to ad see witches are constant integration. Remember that we're substituting x zero unwise to more solving for C so literally just substitute in our values To get rid of the variables and solve for C, we have zero equal see substitute sequel zero back in Now we know we're not done. Remember, the game plan is we need to get why equals? Which means we have to raise are based e We're raising the power to the base E on each equivalent side. So we have Y minus one is e to the X squared over to remember each. The Ellen is simply one. It just cancels. Now there's one last thing we have to do. Remember how the problem initially gave this in terms of F of X? Well, what this means is that we need to substitute why with FX, and then we will be flashed.