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Dy Consider the differential equation 31 2y drLet y = 9(r) be a solution to the differential equatlon with the Initial condition k where k is & constant Euler&#...

Question

Dy Consider the differential equation 31 2y drLet y = 9(r) be a solution to the differential equatlon with the Initial condition k where k is & constant Euler's method, starting at 0 wlth a step 9(0) size of gives the approximation 9(2) ~ 4.5Find the value of k

dy Consider the differential equation 31 2y dr Let y = 9(r) be a solution to the differential equatlon with the Initial condition k where k is & constant Euler's method, starting at 0 wlth a step 9(0) size of gives the approximation 9(2) ~ 4.5 Find the value of k



Answers

Find the solution of the differential equation $d y / d t=k y, k$ a constant, that satisfies the given conditions. $$ k=1.5, \quad y(0)=100 $$

We're interested in this differential equation. Why Prime equals toe? Why? Using all this method with hitch equals 2.2, which equals 2.1. To approximate the values and issue of these X values here for his she goes to zero point to the extra use you're interested in is 0.2 point 4.6 point eight and one initial value is three. So using oil this formula, you find y n from why and minus one plus hitch terms the differential equation value at end minus one. In this case, the differential value here is why. So we'll take why and minus one for example, why one is equals to 10 plus hitch times 10 So these are the values here. You can see the formulas here. So these are the values when the hey cheese point to when he she's 0.1 off course. You need double the number of steps here. So this is the initial value of Y zero. This is the value at when X equals 2.2. So on and so forth. So the table here summarize the final answers thesis the exact why value when Mexico's 20 This is the Is that why value when exposed to point to just keen on Formula three times exponential to the power or whatever X value this year? So as you can see, this is the result that's generated.

Alright for this. Problem is to go ahead and start off by replacing these white terms with our terms. So why does the prime is going to become R squared? Why prime becomes are and why it becomes one? And now we have to do a little bit of factoring and I don't see any obvious factoring. So I'm going to go ahead and use the quadratic formula instead. So we'll have our equals two negative B plus or minus square root of B squared minus four times A times C. All divided by two times A. So that's some fires down a nine plus or minus the square root of 81 minus 36 Which is 45. And that's divided by two and we'll have nine plus or minus while we can write route 45 as three Route five. This is divided by two. All right. And so with this we can actually go ahead and build our solution. So our solution is going to be why equals to see one co sign of three Route 5 divided by two x. Let's see to sign uh huh of three rep 5 divided by two times X. All times E. to the 9/2s X. So that's our answer the differential equation.

Welcome to another differential equation problem. And this one we have nine Y double prime plus nine Y prime minus four Y equals zero. Now to solve this, I'm going to first divide by nine. This will give us a uh first coefficient one and that makes the whole problem a lot easier. I'm gonna rewrite this instead as why double prime plus Y prime minus four. Nineths Y is equal to zero. Great. The next step is to transform this into a characteristic equation and this will happened by r squared plus are when it's four nights, Y equals sorry, last 4/9 equals zero. Great. The next step is to complete the square. Um since we have a one in front of this, our term here, the best solution is to just do our plus one half quantity squared. And if we realize this is going to add a 1/4 term, so we're going to subtract the 1/4 term. Remember we still have this minus 4/9 term here as well? Great. That is equal to zero. The next step I'm going to do is just add all these constants to the right side. So this one and this one and let's look have them let's make them have the same denominator. So we'll make this our plus one half squared equals four nights times 4/4 plus 1/4 times 9/9. Great. And what does that get us? We'll be 16 plus nine. That's 25 over 36. Alright, lucky enough that those are two squares on top of each other. So if we take the spirit of it we'll have our plus one half equals plus or minus 5/6. And if we um subtract one half from this we'll have our is equal to negative one half plus war minus +56 Great. So those are our that's our value of our um let's use our definition of this characteristic equation which gives us solutions the form Y. Equals eat the R. T. To plug in this our value and um get the solution to the difference equation. So I'm going to run up there. Yes I want to just first uh take the positive route of this the negative one half plus five six. So um let's let's write out what are someone is so our someone is equal to um negative three or six plus five or six which is equal to 2/6 or one third one third. Now the second part or two that's negative 3/6 minus 5/6. That will get us negative 8/6 or negative four thirds. Great. Let's plug in these values into our um into our known solution over here And so we have one third negative four thirds. So we'll have uh why is equal to C. One E. To the 1 13 T. Over three plus C. Two E. To the negative four thirds, too, for three. I apologize for the handwriting there, but this is negative 40/3. Great. Um and the reason I'm able to do this is because uh every solution of this form will become a linearly independent uh solution. So if we have a linear combination of these will have all possible solutions for this differential equation. It just happened to be two or three and negative for two or three and that concludes our problem.

For this problem, we need to first find the auxiliary equation. So that's gonna VP of r is equal to on. Then we just replaced the deep it Ours are squared plus nine cubed and said that equal to zero. So this cube indicates that the solutions of this here are going to Hamilton City three. So let's solve this. We're gonna get R squared is equal to negative nine. So that means our is going to be plus or minus three I and then each of these is gonna have a multiplicity of three. So when we're writing, our general solution prefers going to start with the multiple city one. So we're gonna do see one, and then, um, a is equal to zero, so eat a zero is just one, and then B is three. So we're gonna have C one sign of X, That's why. Sign of three x plus, then see too cosigned of three X. Then we're gonna add are multiplicity to so plus C three. That means we're just gonna multiply an axe in front and then sign of three x plus C four because my ex cosigned three x I mean for our last multiplicity. Um, we'll add a C five than an X squared sign of three X and then plus a C six X squared co sign of three X So this is going to be our general solution here.


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