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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
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

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Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.1(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?

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.1That 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

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.1I 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

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.1@amistre64 @freckles

freckles
 one year ago
Best ResponseYou've already chosen the best response.3\[t>0 \text{ since we have } \ln(t) \\ t \neq 1 \text{ since we have } \frac{1}{\ln(t)} \\ t \neq n \pi , n \in \mathbb{Z} \text{ since we have } \cot(t)=\frac{\cos(t)}{\sin(t)}\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.3so we have the intervals: (0,1) (1,pi) (pi,2pi) (2pi,3pi) ,... (npi,(n+1)pi) but t=2 is in (1,pi)

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.1Ahh, ok I see, so it's where they share an intersection, so it doesn't really matter if b(t) is set up as \[\frac{ \cot(t) }{ \ln(t) }\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.3well we already took care of the 1/ln(t) part so we just had to look at the cot(t) part for b(t)

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.1Ok gotcha, thanks!

freckles
 one year ago
Best ResponseYou've already chosen the best response.3but 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

freckles
 one year ago
Best ResponseYou've already chosen the best response.3which was (0,1) U (1,pi) U (pi,2pi) U (2pi,3pi) U ....

freckles
 one year ago
Best ResponseYou've already chosen the best response.3and from that I just looked for the interval that contained t=2

freckles
 one year ago
Best ResponseYou've already chosen the best response.3http://tutorial.math.lamar.edu/Classes/DE/IoV.aspx here are some more examples

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.1Thanks a lot, I think I'm starting to get it now, thanks for the link!
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