## Astrophysics one year ago @ganeshie8

1. Astrophysics

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 $(4+t^2)\frac{ dy }{ dt }+2y=\frac{ d }{ dt }[(4+t^2)y]$ $\frac{ d }{ dt } [(4+t^2)y]=4t$

2. Astrophysics

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

3. Empty

Calculate the derivative on the right hand side of that

4. ganeshie8

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

5. ganeshie8

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

6. Astrophysics

OOOOOH

7. Empty

Yeah, when in doubt, work it out.

8. Astrophysics

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

9. ganeshie8

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.. )

10. Astrophysics

I see I see, thanks!

11. Astrophysics

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

12. ganeshie8

** realized up to a constant term

13. Astrophysics

This is so amazing, omg

14. Empty

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

15. ganeshie8

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 : $y(t) = \dfrac{\int (e^{\int (g/f)\, dt }h/f)dt}{e^{\int (g/f)\, dt }}$

16. ganeshie8

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

17. Astrophysics

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 -.-...

18. Astrophysics

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

19. Astrophysics

I feel I should've known this

20. Astrophysics

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

21. Astrophysics

This is so Newtonian!!

22. ganeshie8

Haha I think it was euler...