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Numerade Educator

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Problem 25 Hard Difficulty

Use the method of Exercise 23 to solve the differential equation.
$ y' + \frac {2}{x}y = \frac {y^3}{x^2} $

Answer

$$y=\pm\left(C x^{4}+\frac{2}{5 x}\right)^{-1 / 2}$$

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Video Transcript

Okay. The first step before solving is to analyze the general form of the equation D y over D backs plus p of axe times. Why is equivalent to queue of acts times in this context wide to the act? This helps us because it's not tells us that p. Of X is to over acts, which means we know we're now making the substitution. You is wide to the one minus on, which is one minus three, which is wired to the negative to which now gives us negative. Why cubed over too do you? Over D Backs is de y over d axe, which means we now have a substitution of d y. Dex could be substituted with negative y cubed over two times D'You over DX multiply each of our terms by negative to over. Why cubed? We simply just flicked what I just said. Our substitution waas so were multiple in each of the terms by this value d'you over D X minus four over acts times one over Why squared is negative to overact squared. Substituting you is one over y squared. We get d'you over D backs minus four over axe you is negative to overact squared. Now remember what I mentioned earlier? Which was what R p of X waas? Well, we have eat the negative four over dried by acts the integral. Thus which is eat the negative four natural of acts which is e to the natural log of X the negative four. Remember, each the natural log is simply one which gives us one over acts the fourth as are integrating factor, we can now multiply each of our terms by this, as you can see I'm doing right now or multiply each of our terms by integrating factor now integrating both sides with respect to X, We end up with the integral of negative two acts to the negative 60 axe which we can write as negative two acts to the negative six plus one. So now we know this is equivalent to one over X the fourth times you simple fire. Just clean this up a little bit. You is one over water squared. These negatives cancel. We have 2/5 two divided by five x. The X could just go on the bottom plus c x to the fourth because remember, were multiplying both sides by sea so we have. See, over here we multiply each of these terms by acts the fourth. Now we have one over why Squared is to over five acts plus C acts the fourth writing this in terms of why again, No exponents just singular. Why is plus or minus? Because again we have the square roots you must take into account that there could be in negative solution. It's imperative to write it with the negative value, and we get our final equation.