Astrophysics one year ago How do you determine an interval in which a solution of the given I.V.P. is certain to exist? @ganeshie8

1. Astrophysics

(Without solving)$y'+\frac{ y }{ \ln(t) } = \frac{ \cot(t) }{ \ln(t) }$ $y(2) = 3$ I see that $a(x) = \frac{ 1 }{ \ln(t) }~~~~~\ln(t) \neq 1$ and $b(x) = \frac{ \cot(t) }{ \ln(t) }$ this is the one I'm having a bit of trouble with, so cot is undefined at pi, and lnt at 1?

2. Astrophysics

That should be a(t) and b(t) so $\frac{ 1 }{ lnt } ~~~t>1$ but b(t) is where I'm having a bit of trouble

3. Astrophysics

I know that 3 has no effect on the interval, and that y(2) we have t=2 so I guess I need to figure out where exactly b(t) is continuous

4. Astrophysics

@amistre64 @freckles

5. freckles

$t>0 \text{ since we have } \ln(t) \\ t \neq 1 \text{ since we have } \frac{1}{\ln(t)} \\ t \neq n \pi , n \in \mathbb{Z} \text{ since we have } \cot(t)=\frac{\cos(t)}{\sin(t)}$

6. freckles

so we have the intervals: (0,1) (1,pi) (pi,2pi) (2pi,3pi) ,... (npi,(n+1)pi) but t=2 is in (1,pi)

7. Astrophysics

Ahh, ok I see, so it's where they share an intersection, so it doesn't really matter if b(t) is set up as $\frac{ \cot(t) }{ \ln(t) }$

8. freckles

well we already took care of the 1/ln(t) part so we just had to look at the cot(t) part for b(t)

9. Astrophysics

Ok gotcha, thanks!

10. freckles

but yeah b(t)=cot(t)/ln(t) is defined and continuous on: (0,1) U (1,pi) U (pi,2pi) U (2pi,3pi) U .... and a(t)=1/ln(t) is defined and continuous on: (0,1) U (1,inf) so yes I just took the intersection

11. freckles

which was (0,1) U (1,pi) U (pi,2pi) U (2pi,3pi) U ....

12. freckles

and from that I just looked for the interval that contained t=2

13. freckles

http://tutorial.math.lamar.edu/Classes/DE/IoV.aspx here are some more examples

14. Astrophysics

Thanks a lot, I think I'm starting to get it now, thanks for the link!

15. freckles

np

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