Question

@ganeshie8

Answer 1

I need help understanding what's going on in this example, \[(4+t^2)\frac{ dy }{ dt }+2ty=4t\] solve the O.D.E then it shows the left side \[(4+t^2)\frac{ dy }{ dt }+2y=\frac{ d }{ dt }[(4+t^2)y]\]
\[\frac{ d }{ dt } [(4+t^2)y]=4t\]

Answer 2

Oops it says \[(4+t^2)\frac{ dy }{ dt }+2ty=\frac{ d }{ dt }[(4+t^2)y]\] forgot the t

Answer 3

Calculate the derivative on the right hand side of that

Answer 4

They are simply using the product rule in reverse : \(f'g+fg' = (fg)'\)

Answer 5

This might be easy to compare against : \(fy' + f'y = (fy)'\)

Answer 6

OOOOOH

Answer 7

Yeah, when in doubt, work it out.

Answer 8

Thanks haha, I think this leads up to the integrating factor, so it's just another of the techniques

Answer 9

Exactly, that integrating factor business is all about making up the left hand side so that it is ready for using product rule in reverse combined with the fundamental theorem of calc( integral is the inverse of derivative realized up to a constant factor.. )

Answer 10

I see I see, thanks!

Answer 11

I've never really used the product rule in reverse, this is nifty

Answer 12

**
realized up to a constant term

Answer 13

This is so amazing, omg

Answer 14

yup it's definitely one of my favorite tricks I wish all ODEs could be solved by integrating factors haha

Answer 15

Basically we can solve "ANY" differential equation of below form using that integrating factor trick :
\[f(t)y' + g(t)y = h(t)\]

the solution is given by :
\[y(t) = \dfrac{\int (e^{\int (g/f)\, dt }h/f)dt}{e^{\int (g/f)\, dt }}\]

Answer 16

It is my fav too, and it really becomes powerful in cases where you want to make a DE exact...

Answer 17

Sweet haha, ok thanks guys, I may have more questions as I read more about ODE xD, as sometimes the book seems to skip steps -.-...

Answer 18

So if I have \[\frac{ d }{ dt }[(4+t^2)y]=4t \implies d[(4+t^2)y]=4tdt \implies (4+t^2)y=2t^2+C\] so the d cancels out the integral

Answer 19

I feel I should've known this

Answer 20

\[(4+t^2)y=\int\limits 4t dt \implies (4+t^2)y=2t^2+C\] to be more clear

Answer 21

This is so Newtonian!!

Answer 22

Haha I think it was euler...