Solving a Differential Equation In Exercises $7-10$ , find the general solution of the differential

equation and check the result by differentiation.

$\frac{d y}{d t}=9 t^{2}$

$$

y=3 t^{3}+C

$$

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huh? We're working on a problem from chapter four, Section one. It's number seven. Directions are to find the general solution of a differential equation in the equation given is do you? Why TT equals nine. T. I swear, when we're all done, we want this to look like why equals and then some sinti stuff. Okay, so that's what we're trying to get to. Um, What we need to do is we just need to integrate both sides of the equation. But we can right now because of this mess here. Um, these differentials are jumbled together, but because they're using a handy live nous notation, we can treat the differentials as if they're numerator and denominator of fraction and weaken separate them. This technique is called separation of Variables. So we basically just have to multiply both sides of the equation by DT the DTS over here. Cancel and we are left with just d y equals nine. See, I swear t t now we are set up so that we can integrates both sides of the equation. The left hand side is very simple, integral of the derivative of why it's just why. But we do need to write plus C because it's an indefinite, integral, the right hand side. We're going to have to use anti power. All recall that power rule says to most by the exploding times the coefficient Then decrease the exponent. We have to do the inverse of that. Okay, We're gonna increase the exponent first. That's gonna be t huge. Then divide the coefficient by this new exponents. So what's nine divided by three? That's three. All right. And the d t goes away. We also have a plus C. Technically, these air different constants perhaps. So C one and C two, but you're going to see a lot of this happened. The Constance just get combined into one constant. Okay, that's exactly what we're gonna do for a last step is basically, subtract this. Subtract this C one over to the other side and join it with C two and Colin, just see final answer. Why equals three t cute. See? The directions also said to check our answer using differentiation. Let me open up a new page and copy down what we have there. What we had I mean, why equals three t cute plus. See? Let's take the derivative. This is implicit differentiation, the derivative of why it's just D Y. It's not one because I didn't say that I'm taking a derivative with respect to anything. It's implicit differentiation. So it's just you. Why now really use power rule again that says to multiply that export at times the corporation just dying three times three is nine and then you decrease the power three decreases, too. But because of the chain rule, we have to write D t. Okay, again, this is implicit differentiation. Now, what's the dream of a constant zero? See, that's what we have to do to plus C because we never know if there was a constant, uh, that may have been there. So our last step here to check and make sure this looks like the original D Y t T equals 90 square. We had to move the d t over by division, basically divide both sides by DT And what are we left with? Very familiar. D y beauty. Oh, my DT equals nine. See square that looks just like the thing on page one. D. Y. D t was 90 square. Very good. So we have checked with differentiation, and that concludes number seven