6 points) Solve the following linear DE with constant coefficients:9" + 44 + 3y = 0.y(0) 0Y(0) = 2...

Find the general solution to the linear differential equation.
$$
y^{\prime \prime}-6 y^{\prime}+9 y=0
$$

So surface problem solving for that module solutions will have the art square six are plus nine equals zero. And this factor's to r plus three square equals to zero. And so we have repeating roots of our equals negative three. So with this we can actually build or homogeneous solution. Homogeneous solution in this case can be C one each. And I have three X plus C. Two X. E. To the negative three X. And we are we add this X term right here because it is repeating. And so now we can actually build our guests for the particular solution. So our guests for the particular solution is going to be of the form A. E. To the negative X. And we want to take the derivative this twice the first derivative is gonna be negative E. To the negative X. And our second derivative is going to be A E. To the negative X. And so we want to plug these these equations right here into the original equation that we had which was wide of a prime plus six. Wide prime plus nine Y equals two. Two eaten. No you have to X. And so Y. Double prime. We said was A to the negative X plus six or other. This could be a minus here because they're multiplying by negative. So we'll have six A. Eat and negative X plus nine A. E. To the negative X. It's all equals two. E. To the negative X. Right. So we can simplify the left hand side and they will give us four A. E. To the negative X. And that equals two E. The negative X. We can cancel these two terms right here. So have that foray equals two. We divide by four. We have the E equals to one half. And so with this we can actually build our total solution which is the sum of her homogeneous solution plus the particular solution. So in this case we said that a homogeneous solution was C one E. To the negative three X. Let's see two X. E. To the negative three X. And our particular solution is going to be one half E. To the negative X.

Because we're going to buy Double Prime. It's five by friends. That's 12. By as you could, you know. Okay, so we're gonna solve this homogeneous equation. So let's start out by your dragging this as a characteristic equation. So we have two on the squared minus five, but it's 12 because he could stop. Well, Kenley, fact it. Us? Yes, we can. We get to ground the plus three times Lambda minus four. But you got to drop. So we have that Lambda is equal to negative three over two. And for so are characteristic equation would be why, uh h of X is equal to see one inch in England. Three over to us. You, too for our each and four X. Okay. Now itself for art. What cures which we don't have to. Our vision is equal to six. So it's our backs. You could six. You know that. Why p of X? It's a constant. This is equal to pain. Plugging this into our differential equation. We get that too. A double prime minus five. A kind man. It's Wally. And because, well, derivative of a constant, you don't go. So we get negative. 12 a six. So a is equal to negative one over two. So are general Swiss years. Why giggle, too? Or homogeneous a Persian? She one each in a minute, three over to X plus c to need Thio More X minus our particular solution, which is this? A which is bigger in Bordeaux.

Okay, give me why. Double time. 12. Why crime? That's nine. Why is he could still be writing this in terms. Wondering for land The squared when it's 12 limber plus nine is you got to go on land is equal to let's use our quadratic formula that swelled plus minus is great root of 1 44 minus four times four times nine all over two times for this gives me 12 plus minus zero over its which gives me on the is equal to three over two. So we get that reaction to out of this, Not the one. Remember to write, since these were too. This was one was plus and better ones minus. So we have a repeat sort equation is why is equal to see one you to the three over to next. Both see two times x times need to dream over too. X

Hello. We have to solve that given defense location that is nine. Do you know why this four way? It cost 20 so we can take the auxiliary cushion. It is an animal script is fought. It cost 20. So amble because to Plus manage to buy three of iota. This is the complex roots. So the solution of the differential equation can be a tennis ball contribute to the power of l. Sandy tax. Plus you do because of the attacks. Okay, so this is alpha zero m is Stephen sign of Beatus two by three. Ex prosciutto of course To buy three of X. So why will be caused to see even off Shine to where three of X plus you 12 Cost two or 3 of X. So this is the answer. I hope you understood. Thank you.