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Solve the differential equation.

$ xy' - 2y = x^2, x > 0 $

$$y=x^{2}(\ln x+C)$$

Differential Equations

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In order to solve this equation, we want to get into the form y prime plus y p of X is Q backs as listed in the textbook and order. To this, we must divide both sides by X to get wide prime minus two. Why over acts is acts. Now we know we must multiply each of these terms by the constant of integration. So each the integral is night of negative to over Axe de Axe gives us eat the negative to natural law of acts which gives us eat the natural of X to the minus two, which is acts to the minus two because again, each the natural log simply gives us once This can literally just be crossed off. Okay, Now that we have this, we know we're gonna be multiplying the integrating factor by each of our subsequent terms. Because then were we able to integrate the right side So you can see we're just multiplying each other charms by the integrating factor. Okay, cool. Now that we've got this, we know the left hand side remains. Why axe to the negative too? However, the right hand side we now need to integrate DEA vacs over acts which is the same thing is integrating one over ox, which is natural log of X plus C. Now, lastly, for a last step, we know we must divide both sides in order to get why by itself. So we're dividing each of these terms by X to the negative too. So we end up with why is X squared times natural log of acts plus see are integrating factor.