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Problem 12 Easy Difficulty

Verify that the following functions are solutions to the given differential equation.
$y=e^{x^{2} / 2}$ solves $y^{\prime}=x y$

Answer

$y^{\prime}=\frac{d}{d x}\left(e^{x^{2} / 2}\right)=e^{x^{2} / 2} \cdot x$
(exponential differentiation, chain rule and general power rule)
$=x\left(e^{x^{2} / 2}\right)=x y$ (satisfying the differential equation)

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Video Transcript

So because of the fact that our differential equation involves a Y prime, the first step in solving this problem is going to be finding the first derivative of why So why prime is going to be equal to We're gonna have to do the chain well here. So, first we'll take the derivative of X squared over two and using the power rule, we know that this is going to be two x to the first power over to and this will be times E to the same exponents, um X squared over two. And so when we simplify, we find that why Prime is equal to X e the X squared over two. And so when we substitute of this into our differential equation, we know that Ah, why prime again is X e to the X squared over too. And this should be equal to X times. Why? And so we know that why is equal to e to the X squared over two. So when we plug all this into our differential equation, we get X E to the X squared over two is equal to x e to the x squared over to power. So since he's a reek willl that tells us that why is a solution to