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Point) Solve the given initial value problem xy y =y 3x3 y(1) = 6y(x)...

Question

Point) Solve the given initial value problem xy y =y 3x3 y(1) = 6y(x)

point) Solve the given initial value problem xy y =y 3x3 y(1) = 6 y(x)



Answers

Solve the initial value problems. $$\frac{d y}{d x}+x y=x, \quad y(0)=-6$$

Okay, so your X squared plus 36 times Why? Over X one. Let's put up our he lies and the exit So we have that g y is equal to one over x squared plus 36 He x knowledge figure in a row of upsides. Okay, well, this is this tensions and rested its why is equal to one over six changes in verse of X over six seat. Now let's plug in our initial value, he said. That's why of six is equal to dough, so we have geology to one over six tendons. Inverse of Joe. What tangent? Inverse of zero that's equal to Oh, no, that's not attended in verse six over six. That's one thinks that it's pi over four. Do it fiver four times, one over six. So that's why, over 24 let's see if he could still sees it was negative. Pi over to what plug in what we know, we said. It's why is equal to one over six tensions. Inverse of X Over six foot sea, which is negative high over 34

Okay, so we have the different equation here and why Double crime plus six x is equal to zero. And then we have that. Why of zero? Basically one. And then why? Prime of zero is equal to two. So we can go ahead. And we can rewrite this difference equation as why step a prime is equal to negative six acts. Okay. And then we integrate both sides twice. So the first time right? Integrating here we get that or the growth of why it double crime is the first derivative. My prime is equal to well in a girl here and give us a negative three x squared plus adultery constant. It's called Seat one. Hated Integrating again. Gives us why is equal to, um, negative X cube. Um, right over 300 canceled. Just get negative X cube. And then we get Well, um, in agro of C one is going to give us plus C one times X. We pick up another constant, uh, see too. Okay, so now we go ahead and reuse our given initial conditions. So we know that why prime of zero equals two. So why primes? Zero. Um, is equal to to give. That implies that two is equal to see one. You know that why of zero is equal to one came off. That implies that one is equal. Ju si two. So therefore we get, um, a solution to our initial value problem. So our solution here is why Ivax is equal. Chew um negative X cube. Plus she wrecks plus one her it

In this problem were given a differential equation over here, and we want to find a solution for it that satisfies these initial values that are given. So to do that, we can go ahead and in a great to get that Why double prime of X is the indefinite integral of six x dx, which is equal Teoh um one Halftime. Six So three x two The two plus C one. Um, that's just using the reverse power role here. So then we can go ahead and integrate both sides again to get why prime of X equal to the integral of three x squared plus C one d x, which is equal Teoh again using the power rule, um will be X to the third plus C one x plus some constant c two And finally, we can integrate one more time to get our solution. Why have x equal to the integral of x? 1/3 plus c one x plus seat to d X equals 1/4 x to the fore plus 1/2 see one x squared plus c two x plus, some constant C three. All right, so now we have all that and we want Teoh plug in these initial values. So this is a general solution to our differential equation. And now we want our solution plugging in our initial values. So we'll start up top with this wide, double prime equal to three x squared plus e one. And we can plug in our condition over here that we want. Why? Double prime of zero equal to four. So why Double prime X equals three x squared plus C one. So we want four equal to three times zero squared plus C one, which means see one is equal to four. All right, we can use this in our second equation. Will look at why prime now and our condition on why Prime? So we'll do that right here. We want why Prime of X is equal to X to the third plus C one of X. Where do you know what C one is? It's four. So we can just write four x plus c two and then we want to plug in our conditions. So why prime of zero? We want to be negative one equal to zero to the third, plus four times zero plus C two so this gives us that si two is equal to negative one and then finally will use this condition that we want y of zero to equal one supposed to be in blue. Why zero equal toe one. So we can use our equation for why here and plug it in. So we have rat Why the vax is equal to 1/4 x to the fourth plus 1/2 See, one is for we know X squared. Plus Now we know C two is negative one so negative one times x plus C three and we want I have zero equals one. So we set one equal to 1/4 0 to the fore, plus 1/2 times four times zero squared plus negative one time zero plus C three All the zeros cancel and we get that one is equal to C three So we can finally plug that in Teoh r Y of X to get a final, um, solution to our problem. So our final solution will be why vax equals 1/4 x to the fourth plus half of fours to two x squared minus X plus one. And this is a solution that satisfies all of our initial conditions and solves our differential equation

All right, Silver problem 31. We have to find a particular solution to this differential equation with the given initial values. So since what? This is a simple differential equation in that older coefficients are constants. We can assume that the solution is in the form of E to the power for constant times, the independent variable. In this case, I'm gonna write us acts. So we're gonna differentiate. Uh, so why problems he goes, okay. Times either Car of chaos and the white double crimes. Here was a case. Great, because the seats of power kickbacks and we substitute these into the differential equations. So it's gonna be case query times either Parquet axe for us. Five times King things, each of our kayaks plus six times either power KXAS because zero, we factor out e to the power of chaos. So it's gonna make each of the parquet x times K squared plus five k from six. It was a zero. We're gonna find the values of K such that this equations he goes a zero. So we're gonna focus on the right part, since the last part is it is an exponential function and doesn't never ecos. It's hero. So for the right part, that conveniently becomes a quadratic equation. So K squared plus five k plus six is he goes to zero, we're gonna factor It says give me Cave was two and K plus three And those are solutions for K are like the three and they go to And just like that, we've We've also found the solutions to the differential equation. So we're gonna write it as why is equal to a constant times e to the power connective three acts, and then we add the other solution lived it. So it's gonna be plus another constant times e to the power of negative two acts. And with this general solution, we're gonna plug in the initial values to find these unknown Constance. So first y zero equals zero. So when x zero lions zero So it's gonna be zeros equals sc of one times e to the power of, well, the exponents going t zero. So it's gonna be you to the ground zero because the C two time seat of have the exponents also gonna need zero. And since that u to the power zero is just one, we can just ignore them. And thus we got zero is equal toe, the first constant plus the second constant. Which means since we have, like, two unknown variables in the one formula, we're going to have to go with the second initial value so before, just a different differentiated since its ally prime and then plug in the values. So if we differentiate ever gonna get blind secrets negative three times a constant times E to the power number three acts, they're minus two times the other constant times. You took power in two acts, and then we plug in the values. So when x zero Why, prime will be negatives here selling them for prime hero since I forgot. So it's gonna be negative. Teoh? Yes, it was number three times CIA One. Well, the exponents give me zero. So it's going to turn to me one lines to time to see two and same for the each of the power of zero. And then we have two formulas to find a to constant. So it's me and 06 C one plus C two. I'm gonna start to see one in secrets and negative C two. So we're gonna like, substituted into the first the equation. So it's give me negative, serious secrets. Um, I go three times, See? One and the since seats here is secret Mega C one, we're going toe substitute the stay here, so it's gonna be for us. It's seen one. So when they go to a sequel, Teoh Negative C one those a C one in secret until which means the seats service being negative too. Since both of these adult together should equal zero. And just like that, we found our particular solution to the differential equation. So it's gonna be why is equal Teoh serious times seeds from things of three acts minus two times eat on the go to lax and, yeah, that's basically adds


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