Question

ODE question: Assume that \(\phi "+p(t)\phi ' +q(t) \phi =0\) admits solutions \(\phi=t\) and \(\phi(t)=e^t\). Find p(t) and q(t). Also solvey"+p(t)y'+q(t)y=f(t).

Answer 1

I am stuck. I found \(p(t)=c-\frac{1}{t-1}\) but how do I find q(t)?

Answer 2

i found p(t) using Abel's theorem. But there is no equivalent I can find for q(t)

Answer 3

Curious did you solve this one since you close it?

Answer 4

yea I found my mistake

Answer 5

I didn't take the derivative of C first time round... it was stupid

Answer 6

I'm not totally familiar with abel...But I was thinking this and maybe it is not appropriate . I know you have the answer but I kinda wanted to give it a shot too. :)
\[ \phi=t \\ \phi'=1 \\ \phi''=0 \\ 0+p(1)+q(t)=0 \\ p+qt=0 \\ \text{ now the other solution } e^t=\phi=\phi'=\phi'' \\ e^t+pe^t +qe^t=0 \\ \\ \text{ now the combination of those solutions } \\ \phi=e^t+t \\ \phi'=e^t +1 \\ \phi''=e^t \\ e^t+p(e^t+1)+q(e^t+t)=0\\ \text{ we sorta have a system to solve now } \\ p=-qt \text{ from first equation } \\ \text{ second equation } e^t+pe^t+qe^t=0 \text{ can be written as } 1+p+q=0 \\ \text{ so } p+q=-1 \\ \text{ so hmmm using first equation with last equation I got we have : } \\ e^t-qt(e^t+1)+q(e^t+t)=0
\\ \text{ so we can solve for } q \text{ I believe } \\ \\ q[-te^t-t+e^t+t]=-e^t \\ \\ q=\frac{-e^t}{e^t(1-t)+0}=\frac{1}{1-t} \\ \text{ then } p+q=-1 \\ p+\frac{1}{1-t}=-1 \\ p(1-t)+1=-1(t-1) , t \neq -1 \\ p(1-t)=1-t-1 \\ p=\frac{-t}{1-t}
\\ \text{ now I didn't consider the most general from of the sum solution I think }


\]

Answer 7

yep that's p(t)

Answer 8

I just plug in the solutions basically and also plug in the sum of the solution into differenitial equation
and solved a system of equations

Answer 9

abel's is just faster, it's at the bottom of this page http://tutorial.math.lamar.edu/Classes/DE/Wronskian.aspx

Answer 10

nice work

Answer 11

see if you can get q(t),it's not too far off from p

Answer 12

yah since their sum is suppose to be -1 :)
Or that is what I have :)

Answer 13

Anyways cute question

Answer 14

hahahaha thanks. It's this "review" sheet for my test. He didn't cover half of it...

Answer 15

I think that is the way that most advance classes go

Answer 16

too much to shove into class lectures

Answer 17

no, he's just a spiteful (insert inappropriate word here)

Answer 18

he is so awful

Answer 19

lol

Answer 20

Well hopefully you will be done with him soon.

Answer 21

check out this question, I still have no answer. No one at my university can do it apparently http://openstudy.com/users/fibonaccichick666#/updates/54e4068ee4b00ad827d6874a

Answer 22

only a month and a half left. I cannot wait