Hi all,
If we have
H'(r)=r+\tau(r)H(r)
and
\tau(r)=k+(H(r)/r)^a
where
a>0, k>0, and H(0)=0,
can we say anything about {lim}_{r\rightarrow 0^+}\tau(r)?
Thanks a lot!
Well, we know that y_1(r)>y_0(r) for all r.
If y is defined for all r, isn't it necessarily piecewise continuous?
k, at any rate, is continuous.
It seems to me that there has to be a neighborhood to the left of r_0 in which y_1(r)-y_0(r)>k(-H_0)H_0-k(-H_1)H_1...
Beautiful solution, once again :)
I was wondering, whether the fact that y is not necessarily continuous matters.
Can I speak about (H_0-H_1)'(r_0) and y_0(r_0)-y_1(r_0)?
Or do I need to deal with (H_0-H_1)'(r\rightarrow r_0^-).
And then, can I claim y_0(\rightarrow...
Thanks a lot! That was a very elegant explanation :)
Now, can I prove the same for:
H'(r)=-y(r)-H(r) k(-H(r)) ,
where k is a strictly increasing positive function?
Hi all,
I'd be very happy if you could help me solve a problem in my research.
I need to prove the following:
H'(r) = -y(r) - k H(r)
k is a constant.
y is strictly increasing, but not continuous.
Let (a,b]\subset R .
(H_x, y_x) denotes solution x.
H_1(a)<H_0(a)<0 ...
Thank you so much for your help!
I can treat them as coupled. The problem is knowing when to switch from one set of coupled DEs to the other.
If I start with w<wm, I'm in the w<wm and H>0 region. This region ends when w>=wm (because H continues to rise). How do I know when this condition...
Hello all,
I am dealing with a rather complicated problem (as we all do), without much knowledge in differential equations.
I have coded a numerical solution, taking a straightforward approach: constructing the functions step by step, but I would like to try to achieve better precision...