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Numerade Educator

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Problem 19 Hard Difficulty

Solve the initial-value problem.
$ xy' = y + x^2 \sin x, y(\pi) = 0 $

Answer

$$C=-1, \text { so } y=-x \cos x-x$$

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Video Transcript

In order to determine the interpreting factor, we must first put it into the standard form of the equation. Ever Words must divide both sides by the coefficient of white prime, which in this context is X because we are exported prime. So again divide all the terms by this value to give us this. Now we know our p of X is negative one over X. Therefore, integrating factor each the negative times integral of one over X D acts gives us eat the negative natural of axe E to the natural log is one which gives us negative one times acts which in this context would be X to the negative one, which, in other words, is the same thing as one over X to the power of one which is just one over X. Therefore, multiply both sides of the differential equation by one over acts. We end up with one over acts. Why is equivalent to the integral of sine of axe de luxe? We end up with writing this in terms of why we know the integral of sine of X is negative co sign X and remember, we want why by Self Smiths divided to the terms by one over X we get Why is negative x co sign acts plus C times acts now are factor off Why have pie is Europe In order to solve for C, we plug in what we have zero is negative pi times co sign of pi plus C times pot As you can see, I'm literally playing this right into the equation we just found we gotta solve for C We underfoot see his native won Our final equation solution is why is negative acts co sign acts minus X.