Question
Point) The system of first order differential equations:J yi y210 y1 + 3 y2, 215 Vi 2 Y2,y1 (0) = 2 Y2 (0) =-9has solution: Y1 (t) y2(t)Mote: You must express the answer in terms of real numbers only:
point) The system of first order differential equations: J yi y2 10 y1 + 3 y2, 215 Vi 2 Y2, y1 (0) = 2 Y2 (0) =-9 has solution: Y1 (t) y2(t) Mote: You must express the answer in terms of real numbers only:
Answers
Find the general solution of the given second-order differential equation. $$y^{\prime \prime}-3 y^{\prime}+2 y=0$$
Okay so start this problem and it go ahead and do a regular substitution. So with that we'll have three R squared plus two are plus one equals to zero. And since I don't see any obvious factor, I'm gonna go ahead and use the quadratic formula so have negative B. Oh plus or minus the square root of B squared -4 Times A. Times C. All divided by two times A. All right. Let's go ahead and simplify this out just a little bit. So we'll have negative too. Plus or minus the square it of let's see that's gonna be 12. So a negative eight divided by six. And we can go ahead and write this as negative two plus or minus the square of two. I wrote to divided by six. Mhm. We can factor out the two over here and cancel that with the denominator. So we'll have negative one third closer minus I root two divided by three. And these are our values in this case and with this our value right here we can go ahead and build our solution. Sir. Solution is going to be the form Y equals two K one E. to the -1 3rd x. Co sign Of Route two divided by three x. Plus K two E. to the -1 3rd x. Sign Of Route two divided by three x. So that's your answer in this case.
Okay, So what we have here is a differential equation Working were asked to get it solutions you. Then its initial review showed, which is off white at zero is equal to 10. So what we want to do first this is is to transfer. Why? Over three to the left side, to the right side. I mean off the pressure. And this will give us the Y over the eggs. Physical negative 1/3. Why next? We removed it by one over. Why the both sides and we will have the wife over. Why the X is equal to negative 1/3. Next. I wonder what they liked the ex to both sides, and this will give us the way over. Why is it called a negative 1/3 X? So integrating both sides of the equation we'll have and in a way is equal to negative 1/3 X. Let's see. We can also express this a ski race to Ellen of why is equal to either use the negative 1/3 eggs. Let's see. So we know that here is the island of why it's just why and see it's just a constant so it can take any values. Therefore, we can express this as C E brings the negative 1/3 x So this is now our generals should So we want to find the particular solution when we do the speck of getting for the value of C. So in order to compete toe compute first see, we make use of the initial conditions given, which is at X is equal to zero. Why is equal to 10? Therefore we substitute this So the general solution we have then is equal to see erased a negative one over B zero. This will just be equal. The one and the value off C is equal to 10 rewriting the clinically shown we have. Why is it well then erased in the negative 1/3 eggs? And this is now our final answer, which is the particular solution off over differential equation.
Okay, so let's go ahead and do our general substitution which is going to be replacing. Why? Double prime with R squared and why with one? So in this case will have three hours squared plus one. He calls zero. I can bring the one to the other side or the right hand side. Rather we'll have three. R squared equals negative one can divide by three. So have R squared equals two negative one third. You can take the square root. So have our equals to plus or minus root negative 1/3. So I can rewrite this as our equals to plus or -1 Of Route 1 3rd. Right. And so with this actually we can build our total solution. There are a total solution in this case it's going to be Y equals to see one co sign route one third. X plus C to sign of one divided by three uh squared of that times X. We can sympathize out a little bit. So if we divide uh or distribute the square roots, the numerator and the denominator. In this case The square root of one is simply just one. And so we can go ahead and rewrite this to be one divided by Route three X. Okay. See to sign of one divided by worth three X. And so that will be your answer.
Okay, so start off this problem. Let's go ahead and remove these wide old prime term and the white the white term. And then we're going to do that is we're going to substitute in our square and one. So I've r squared minus 36 equals to zero. And we can do a little bit of factoring here. They should factor down to AR -6 times are plus six That equals to zero. And from this we can derive that are has to equal -6 and six. And so with that information we're able to build a homogeneous solution. And since in this case we don't have anything on the right hand side or homogeneous solution is equal to our total solution. Our total solutions can be K one, eat the negative six X plus K two E to the six x. And so that's gonna be your final answer.