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Problem 22 Hard Difficulty

Solve the differential equation $ xy' = y + xe^{y/x} $ by making the change of variable $ v = y/x. $

Answer

$y=-x \ln (C-\ln |x|)$

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WI

What I.

July 31, 2021

Numerade, can we please get someone to actually explain how to do this instead of someone speedrunning the questions to pad their ExtraCurriculars

Video Transcript

this question asks us to solve the differential equation. X Y prime is why, plus accede to the power of why over acts by using the fact that V is why over X now What this means is that, given the fact that V is, why over acts, right, this just in terms of why, as why is X times V not differentiate, then divide by detox. Now we know we're going to look at the given differential equation, divide the equation by acts and then substitute. We have X TV over detox is eat the Power V, not separate variables and integrate. We can separate variables right now with old Avi terms on the left hand side and all the ex terms on the right hand side before we integrate. Let's write this with negative powers, because the fractions make it a little complicated when integrating the integral of the integral sign has been added to both sides. Now, when we integrate, we get negative E to the negative V is natural log of acts plus two c Recall that Caesar constant of integration. Now we have negative y over oxen left hand side, and the reason why is because if we multiply both side by natural log, we know that the natural log times X or not formed the power start natural logs that power e natural like times E simply cancels out and is equivalent to one. That's why the ease disappeared from the left hand side.