Question

Differential Equation Answer Check... help likely needed as well.

Find the general solution of the given differential equation, and use it to determine how solutions behave as \(t\rightarrow \infty\)

\(2y\prime+y=3t^2\)

Answer 1

The answer that I arrived at is:

\(y(t)=e^{\frac{-1}{2}t}(3e^{\frac{t}{2}}(t^2-4t+8))+Ce^{\frac{-1}{2}t}\)

However, I feel like this should be able to be reduced somehow.

Answer 2

Yes, you could still simplyify it a bit more to make it look neater. Lol

Answer 3

How on earth.... I feel like such a schmuck for asking so many questions that seem so simple.
How would one set about simplifying this?

Answer 4

distribute the 3e\(^{\frac{t}{2}}\)...then the e\(^{-\frac{t}{2}}\)

Answer 5

Wait...
Would the \(e^{\frac{-t}{2}}\)cancel out the \(e^{\frac{t}{2}}\), and then Leave just the 3 and that would distribute?

Answer 6

You \(\sf \color{red}{don't}\) NEED to simplyify tho, this isn't an algebra course, BUT just easier to read if you're doing this on an exam.

Answer 7

I was just saying you could of simplified it a bit for me! lol.

Answer 8

\(y(t)=3t^2-12t+24+Ce^{\frac{-t}{2}}\)

Answer 9

yes.

Answer 10

You shouldn't be struggling on simplification tho at this level of math.

Answer 11

something wrong above , please check. where is your int?

Answer 12

I am having a small series of brain farts ok?
And, what is wrong??

Answer 13

I think he did his work on his paper. There's no need to show integral here. The integral is one of the basic integrals and is able to be integrated.

Answer 14

because some answers are not integrable and must be left as an integral at this level of mathematics.

Answer 15

I know that, I have some of those in my notes. However, this one is integrable, and as such isn't a necessary part of the answer right?

Answer 16

to me, \[y = e^{-\frac{1}{2}t}\int 3t^2e^{\frac{1}{2}t} + C e^{\frac{1}{2}t}\]

Answer 17

and for the int. you have to use tabular to solve . I mean integral by part twice

Answer 18

Ok.

Answer 19

cannot stay, my "trash " computer doesn't work well

Answer 20

Which ever method he used, he arrived at the correct solution. Plus, I am not his grader so I have no idea whether he was doing it correctly to give him "full credit". So I have no say in his methodology.

Answer 21

Ok, well... I have to continue working out problems. Should I keep posting them here? Or shall I start new questions each time? I don't quite feel confident yet.

Answer 22

he could of used laplace, or series to solve this, or used the integrating factor method.

Answer 23

@abb0t . I am new in this field. I have to be careful

Answer 24

It's ok, Loser. If you have questions, post them up. There are a lot of users on here who can help in ODE's. I am probably not that person. But i can recommend some who know this math course.

Answer 25

And to clarify for you @abb0t we are coving integrating factors right now, so I am to assume that is the method he would expect us to use.

Answer 26

And to clarify for you @abb0t we are coving integrating factors right now, so I am to assume that is the method he would expect us to use.

Answer 27

Ok.

Answer 28

My next problem is an Initial Value Problem.

\(y\prime -y=2t~e^{2t}\)
\(y(0)=1\)

\(y(t)=\dfrac{2(3t-1)e^{2t}}{9}+Ce^t\)

Then you would plug in zero for t, and then solve for C right?

Answer 29

Anyone have any input?

Answer 30

I don't check your stuff, just answer your question. Yes,

Answer 31

Then, do you plug what you have back in for C, and then that is your final answer?

Answer 32

Again, Yes

Answer 33

Are you solving physics part? find velocity? distance? or spring mass?

Answer 34

Nope, it is just a plain jane initial value problem.

Answer 35

ok, good luck.

Answer 36

Thanks, I think I have it!