Right. So our initial value problem is going to be the third derivative of why. Plus two. Why two times the second derivative, minus nine times E first derivative minus 18. Why is equal to negative 18 X squared, minus 18 eggs plus 22. So the corresponding auxiliary equation is our cubed plus two R squared minus nine. AR minus 18 is and we set that equal to zero. If we factor it, it's gonna be the same as AR minus three times R plus two times our first three being equal to zero. So that means that are ours are equal to three negative to negative three. So the homogeneous pushing of the solution is some constant times e to the negative three x plus some other constant times e to the negative two x plus some third constant times e to the power of three x So now we want to find the particulate a part of the solution and that will be in the form X squared plus B x plus c. So the first trip did of that is gonna be to a X plus B thesis. Aiken, derivative of our particular solution will be to a and the third derivative of the particular solution zero. I'm gonna put all these into this original. Ah, problem and we would end up with is zero plus for a minus 18. A X minus nine. Be minus 18. A X squared minus 18. Be eggs minus 18. See which is equivalent to was gonna put things. Ah, instead order. So a negative. 18 a x square plus negative 18 a minus 18 B, times x plus for a minus nine B minus 18. See? Okay. And so that we wanted to accept this equal to what was on the right hand side of that initial value problem? Yes. So this is going to be equal to negative 18 X squared, minus 18 X plus 22. Okay, well, so we want to compare the coefficients. So we have, um negative 18. A needs to equal negative. 18. So that would imply that a is equal to one. Came in for the second one. We have negative. 18 A. We know that a zygote a one. So the night of 18 times one minus 18 times be needs to equal negative 18. Okay, so that means negative 18 minus 18 b Mystical native 18. So this surprise that b is equal to zero their last one four times a just one minus nine times zero, which is zero um minus 18 c needs to equal 22. I guess this means native 18 c is equal to 18 so C is equal to negative one. So then we have, uh, the pieces we need for our particular solution. And that means that the particular solution is equal to is gonna be a X squared is one such as X squared plus B x p zero ah plus C minus one. So x squared minus one is the particular solution. And so if we add our particular solution to the Virginias portion, we get that the full solution is equal to x squared minus one plus some constant times e to the negative three x plus some other constant times e to the negative to eggs, plus some third constant times e the three x. Okay, so this is really our general solution and now we want to start using our initial conditions. Now, one thing to note is that the first derivative here is equal to two x minus three times. See one times e to the negative three x minus two times C two times E to the negative two x plus three times C three times either three X, and the second derivative is equal to two plus nine c one e to the negative three x plus four c two e to the negative two x plus nine C three. Either the three x It's not gonna put our initial conditions into these equations. For example, we get why zero, which is gonna be negative one plus C one plus C two plus C three is equal to negative, too. So that means that C one plus C two plus C three must be equal to negative one from the first derivative from why primal zero. We get negative three c one minus two c two plus three C three is equal to native eight and Leslie from the second riveted at zero. We get two plus nine c one plus four c two plus nine C three is equal to negative 12. At least try to from both sides and get that Mayan times C one plus four times C two plus nine times C three is equal to negative. 14 Your hands And then I saw this system of equations. So maybe I'll start by numbering them. Let's just say this one, two and three. So let's start by taking three times the first one and adding it to the second equation. And so the sea ones will get knocked out so we'll end up with See to Plus six. C three is equal to when we take three times negative one and added to the negative eight. So it's gonna equal negative 11. And then we can also take three times the second equation and add it to the third one. Yes, In that case, we're gonna end up with see one being eliminated again s and we have negative six c two plus four c two says the negative to see to it that we have a nine c three plus nine c three so close 18 c three asking equal the three times negative eat minus 14. It's going negative. 38 It is. So that's really leads to new ones as our equation. Four or five. So I want to take two times for equation for and and that to equation five. So the see twos will cancel out. It's gonna give us 30 times. C three is equal to negative 60. So I mean, C three is equal to negative too. But I think I'm just gonna go backwards and put that into another equation. So we can take C three is equal, negative two, And put that into the equation for so you would get C two plus six times Negative, too. So guests minus 12 is equal to negative 11. If we had 12 to both signs, we would get see you choose equal to one. And then we can put both of those into any of our first equations. First three that handle through my vote for C one plus C two plus C three is equal to negative one. So you would get C one plus c to just with one plus c three my eyes to people, the negative one on this means that C one is equal to zero so that we have the values for these constants. We can put that into our general solution we have here. Okay? And when we replace those Constance, our final answer is one of X is equal to X squared minus one plus e to the negative two x minus to E to three x and that will solve our initial value problem.