Sec 2.3: Problem 10Previous ProblemProblem ListNext Problempoint) The differential equation102 63 d 648y U has characteristic equation = 0 help (formulaswith roots ...

For the following problems, find the general solution to the differential equation.
$y^{\prime}=3 x+e^{x}$

The problem AIDS, We have this differential equation. Sorry. Tio Dio already expressed its. That's why. Must sleep was there. We want to transfer it into the standard form off the Nina first order differential equation So we can relax. Worry beside the Y over the axe. Close two acts. Why? Applause? Exercise every you go syrup you can write equals an active X over three. It was negative X over thrift. So our PX is too X hands integrated Factor is it was integral to X t X which is Ito The Axe square. Ah, now we must buy exposed side. Do you lie over the axe? My body to relax square plus two acts One of the e Codex square. Why? It was negative. X over free because the square not this far It is a part of derivative off eggs off this function it was X square line equals Okay, my point is the same. Take it in tow. He did act square. Why? Course now he to the square to action Elective won over six. They multiply to exit guard neck to x o ver sary. Yes, that's correct. Plus, Lee, he wants his part. Why was negative one over six Plast C but by it of connective ex school that set for problem eight

Yeah in this problem we are given the differential equation um that Israel has three X squared uh minus two X plus one. And the first thing I would do is just figure out what anti derivative would work. So as I'm thinking about adding one to the exponents and then dividing by that new exponent um you know three, divide by three will just cancel. And it makes sense because the drift of of execute is three X squared. Same thing with the next one, add one to the exponents. If you divide by that new exponent it makes sense because the drift of X squared is two X. And then the drift of of excess one. But then I'm going to put a plus C in here because then they give you the initial condition that the ordered pair to one works. What does that mean? Is that when X is too so too cute to be eight to square to before plus two plus C, you're going to get a Y value of one. So as I'm going to simplify that 8 -4 is four plus 2 is six. Let's see what equal one. So they subtract six over. The C value must be negative five. They have to go back up here and now the exact equation will be X cubed minus X squared plus X. And now I can replace the C in with negative five and this is the only derivative or only equation that is both the derivative and satisfies that boundary condition.

Question. We are fighting a general solution for this or the so we put it in standard form. We have that p of X is minus two So integrating factor is e to the minus two X and we must fly True, you're gonna have Why times integrating factor off it prime equals three eggs e to the minus two eggs And so we want to integrate the rice I write to fire Why? Let's do this. We pulled three ofus and use into gold bipod as always you will be x the we will be need to minus two x d eggs So our the you will be just the X and we gonna be minus the same function over too. Now we put it in you We will be ex Thanh the exponential or what to my last time minus will become plus integrated e to the minus two X over too d x And this is just exponential and we can duty integration right away. There's going to be home minus e to minus two eggs over four. Yes, And don't forget the car sent to So we have integrated the right hands. I it will be this times tree trying times e to all this. Um, go back here. We have that. Why? Times need Thio minus two. X equals minus tree X over too. Minus tree over full. This to turn times I need to. The mine has two eggs, plus constant o time. Also, Egypt in my nest to x. Sorry, there is no, There's no e. There's no function there. It's just constant. Er Okay, so when we remove this, it's gonna cue this one. But he's gonna add e to the two X over there, and that is the answer. Let me rewrite so minus trio to x minus three or four plus c A cause, then time e power to its And that is Thea Answer. Thank you.

So a section 4.5 problem to 41. We're dealing with a linear, ordinary first order differential equation, and we have a boundary condition. 50 is equal to three on. So what we need to do is just to put this in standard form. And we say, Well, it's already in standard forms. A standard form is why prime some function of X times. Why is equal to some other function of X so I could find the integrating factor is going to be e and so the coefficient of y just one. So this is going to be e to the X is the integrating factor. So that tells me I can rewrite this differential equation as why into the X prime is equal to X e to the X, Then the solution would be to integrate both sides of this equation. So in integrating both sides of this equation, I end up with why e to the X is equal to either the ex text minus one plus a constant of integration then to solve for why it's just a divide by either the x o why is equal to X minus one plus c e to the minus X. Then you have the boundary condition that why of zero 2nd 3 that tells me that to solve for C, I'm gonna have a three. It's a people to zero minus one. See? And then each of the zeros just simply one. So that tells me that C is equal to four. So the final answer here is why is equal to X minus? One was for E to the minus X, and that is the exact solution to this differential question.