Question

Find the solution of each of the following differential equations for a function x = x(t):

dx/dt − 2(x/t) = t^2 − 1 (Hint: Find an integrating factor)

Your assistance is appreciated. Kind regards

Answer 1

My work so far:

1) integrating factor , using -2/t is : e^integral(-2/t dt

2) e^integral(-2/t dt = e^-2lnt = -2t

hence integrating factor is -2/t , or is it -2/t

Correct so far?

Answer 2

\[ \Large \frac{dy}{dt}+ay=g(t) \\
\Large \mu(t)=\exp\left(\int adt\right) \]

Answer 3

Pardon me, I am writing this down for myself basically to remember things, but you agree with this so far?

Answer 4

just one second,,, yes

Answer 5

yup, that's correct

Answer 6

So if you do some pattern matching it seems to me like you have to do the following

\[ \Large \mu(t) =\exp \left( -\int 2tdt\right)\]

Answer 7

is that 2t or 2/t

Answer 8

It is 2t here, that's how I would have done it anyway.

Answer 9

agree @TuringTest ?

Answer 10

Sorry , the question was messed up when I copied and pasted it. That shoul be -2(x/t)

Answer 11

-2t yes'

Answer 12

oh, well...

Answer 13

ohh well that changes things hehe.

Answer 14

I will edit the question...my apologies

Answer 15

then let's rewrite the question
it's okay, it happens

Answer 16

hehe it certainly does hehe

Answer 17

fixed!

Answer 18

\[\Large \frac{dx}{dt}- \left( \frac{2}{t}\right)x=t^2-1 \]

Answer 19

so your integration factor is correct

Answer 20

yep

Answer 21

which one is correct is that 2t or 2/t

Answer 22

as you said
2) e^integral(-2/t dt = e^-2lnt = -2t
that's your IF

Answer 23

\[ \Large \exp\left(-2\int \frac{1}{t}dt=-2\ln(t) \right) \\ \Large e^{-\ln t 2}=-2t\]

Answer 24

oh your last par was wrong

Answer 25

okay....I was getting -2/t from wolframalpha , so i got confused, because my own work gives me 2t...

Answer 26

\[e^{-2\ln t}=e^{\ln(t^{_-2})}=t^{-2}\]

Answer 27

yes I just noticed that I was making a bad substitution there, my bad.

Answer 28

funny how you guys both made the same mistake though :p

Answer 29

I should have written it out with exponentials (-:

Answer 30

lol
so the correct result is t^-2

Answer 31

yes

Answer 32

..for the integrating factor

Answer 33

thanks....I will attempt the solution by myself in a bit ...If I need any help I will give you gentlemen a shout...:)

Answer 34

thank you so much

Answer 35

welcome!