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Solve the second-order equation $ xy'' + 2y' = 12x^2 $ by making the substitution $ u = y'. $

$$

y=x^{3}-\frac{C_{1}}{x}+C_{2}

$$

Differential Equations

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In order to solve this, the first step would be substitute. And you is why prime do y over the X. Because now we have axe d'you over D axe. Plus to you is 12 ax squared get do you over the expert self divide each of the terms by X we end up with D you divide by de ax plus two over acts times you is simply 12 acts because we know this cancels out one of the excess now or integrating factor each. The intro off two of her ex d X is the same thing as eat the two natural of acts which is equivalent to eat the natural of X squared. It's the natural order is simply one which gives us ex quarters are integrating factor Multiply each of this Each of the terms by are integrating factor, which means we now have ax squared of you is three extra fourth plus c. Now we know it's imperative that we end up with you by itself because then we're gonna be back substituting and obviously because we don't have you in our answer, So we're gonna be dividing each the terms by Ark squared to get the war over. Detox is three x squared plus c x The negative, too, Which gives us why is X cubed might see access the night of one plus to see Now these juices are actually different. This is sea of one. This is see if two they're two different interning factors because we entreated the integral of three x squared plus Steve one times x a negative to Jax. We got a second see. So it's important to recognize this and the problem cause now we know we can write our final solution without any negative exponents as X Cube modesty of one over acts plus C of two.