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Given that y_1(t)=t^2 is a solution to the differentialequation: 5t^2*y'' + 5t*y' -20*y = 0, use the reduction of ordersmethod to find another soluti...

Question

Given that y_1(t)=t^2 is a solution to the differentialequation: 5t^2*y'' + 5t*y' -20*y = 0, use the reduction of ordersmethod to find another solution, y_2 using the initial valuesy_2(1)= -1/4 and y_2'(1) = 1/2.

Given that y_1(t)=t^2 is a solution to the differential equation: 5t^2*y'' + 5t*y' -20*y = 0, use the reduction of orders method to find another solution, y_2 using the initial values y_2(1)= -1/4 and y_2'(1) = 1/2.



Answers

Find the general solution of the given second-order differential equation. $$12 y^{\prime \prime}-5 y^{\prime}-2 y=0$$

In this problem, we have to solve the differential equation. Why? Double prank minus four. White crane plus 13. Why is it called to zero? Now we can associate the characteristic equation, which is going to be D square minus fourth Dynasty plus 13 is equal to zero. Now let's try to solve this quadratic equation. So T is going to be negative. Be so negative. Off negative four plus minus Squire rode off B square, so negative force Squire minus for a seat four times one times 30 divided. Fight two times a day. So two times one. So this is going to be four plus minus square root off 16 minus 52 divided by two which is going to be four plus minus. Squire rode off Negative 36 divided by two. So finally I will end up getting too plus minus three heart iota What I owed a negative one. So now we see that the roots Salter characteristic equivalent are complex numbers in this case. So let's recall that you're in. So if our one is it called too Alfa. Plus I'm Peter and are too. Is he called to Alfa minus Iveta Art, though routes off the characteristic excavation characteristic equation. So then we know the solutions are going to be you to the poor Al flicks Time See one times call sign Beata X plus C two Time Sign off B takes seeing our kids. We see that our Alfa is a call to to and our beater Is it going to treat? So then in our kiss, the general solution is going to be, he told the power. Two weeks time. See one time school sign off three X plus C two, then sign off three x So that's going to be oh, general solution for the differential equation.

All right. So I started this problem by replacing the Y double prime term, the UAE prime term. Um And the white term, sorry, left that out here with r squared r and one respectively. So have r squared minus 10 are plus 25 equals zero. Yeah, so we can do a little bit of factoring and we should get our minus five times ar minus five or r minus five whole square. And that equals to zero. And from this we can actually drive the roots which is going to be five and five. Or we can just write it as five. Let's say that this is going to be real and repeating. And with this information right now, what we can do is we can build our solution. So the solution is going to have to form K one X E to the five x plus K two mm to the five X. And note that I added this x term right here simply because it's a repeating route. Yeah. And so this is gonna be our final 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.

Okay, so start of this problem by replacing the Y double prime with R squared the Y. Prime with R. And the Y with one. So we'll have our squared plus for our -1 equals zero. And so I'm doing doing this we can actually factor but since I don't see any obvious factory, I'm going to go ahead and use the quadratic formula. It's negative B plus or minus A squared B squared, which is 16 -4 times a. Which is one time. See all divided by two times A. And so if we do a little bit more simplification here, We can have negative four plus or minus the square of well, yeah, 16 Plus four is 20. All right. and two times 1 is two. Okay, So have negative four plus or minus or we can rewrite the square root of 20. That's two times 10 and two times 5. So we can be rated as to Route five, It's divided by two. And then finally let's go ahead and divide this top the numerator by two. So we'll have negative too plus or minus the square root of five. So that's our our values in this case. And with that we can actually build our Solution so solutions can form C1, Eat a negative 2 -15 X. I'm gonna go ahead and put this in parentheses to indicate that the X. Is going to distribute to both terms. And we also have plus E to the negative two plus route five times X. And so this is her answer.


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