A domain of a solution and a stable of a solution.

D

daniel8851

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A. Find all the solutions.
B. For which values of y0 does the solution defined on the entire real line not exist? What is the maximum domain in which a solution exists in this case?
C. For what values of y0 does more than one solution pass through the point (0,y0)? Write down all the solutions that go through it in this case.
D. Each point y0 through which a fixed solution passes is called a fixed point. Find the setpoints and determine whether they are stable/asymptotically stable solutions.

Hello everyone,
I need some help with this question. And I would also be happy if someone could tell me if my answers are correct. I could answer A, B, and C, but I'm unsure I understood B.
for A the solutions were :y1=0 , y2=-1, y3=(Ce^(2x/5) +1)^(5/2).I omitted y=1 as a solution because for c=0 we get that one. Is that correct?
And for C I got that only for y0=0 2 solutions pass through that point which are y1 and y3.
For B I'm not sure if they mean the solution is defined on all R or that the solution would be correct for all y0.
I struggled to do D, so I would be glad if someone could help me.
Posting when I've done so far.
Thanks, everybody.
 

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daniel8851 said:
View attachment 38254
A. Find all the solutions.
B. For which values of y0 does the solution defined on the entire real line not exist? What is the maximum domain in which a solution exists in this case?
C. For what values of y0 does more than one solution pass through the point (0,y0)? Write down all the solutions that go through it in this case.
D. Each point y0 through which a fixed solution passes is called a fixed point. Find the setpoints and determine whether they are stable/asymptotically stable solutions.

Hello everyone,
I need some help with this question. And I would also be happy if someone could tell me if my answers are correct. I could answer A, B, and C, but I'm unsure I understood B.
for A the solutions were :y1=0 , y2=-1, y3=(Ce^(2x/5) +1)^(5/2).I omitted y=1 as a solution because for c=0 we get that one. Is that correct?
And for C I got that only for y0=0 2 solutions pass through that point which are y1 and y3.
For B I'm not sure if they mean the solution is defined on all R or that the solution would be correct for all y0.
I struggled to do D, so I would be glad if someone could help me.
Posting when I've done so far.
Thanks, everybody.
Click to expand...
For A.

What about [imath]y = 1[/imath]? Do you think that it is not a solution?
 
daniel8851 said:
for A the solutions were :y1=0 , y2=-1, y3=(Ce^(2x/5) +1)^(5/2).I omitted y=1 as a solution because for c=0 we get that one. Is that correct?
Click to expand...
Looks correct to me.
daniel8851 said:
For B I'm not sure if they mean the solution is defined on all R or that the solution would be correct for all y0.
Click to expand...
You have something raised to the 5/2 power -- what are the limitations on that "something"?
 
daniel8851 said:
D. Each point y0 through which a fixed solution passes is called a fixed point. Find the setpoints and determine whether they are stable/asymptotically stable solutions.
Click to expand...
Is "setpoint" the same as fixed point?
Have you learned any stability criteria, like Lyapunov functions, or only the definitions?
 
For B.

I checked some values of [imath]y_0[/imath] but I could not find any values that will let the domain vanish (not exist).

@blamocur
@daniel8851

Has anyone of you found some?

And for maximum domain, is the domain [imath]-\infty < x < \infty[/imath]? Or I am doing something wrong?
 
mario99 said:
I am not really convinced that we should omit [imath]y = 1[/imath] because of [imath]c = 0[/imath]. Since the question asked for all solutions, I would include [imath]y = 1[/imath].
Click to expand...
y3 = 1 when C=0 in the op -- why would you list it separately?
 
mario99 said:
but I could not find any values that will let the domain vanish (not exist).
Click to expand...
The op asks "B. For which values of y0 does the solution defined on the entire real line not exist?". For example, y3 is defined on all of [imath]\mathbb R[/imath] when [imath]C \geq 0[/imath], but for other values of [imath]C[/imath] I am not so sure (post #5).
 
blamocur said:
y3 = 1 when C=0 in the op -- why would you list it separately?
Click to expand...
I see it now.

blamocur said:
The op asks "B. For which values of y0 does the solution defined on the entire real line not exist?". For example, y3 is defined on all of [imath]\mathbb R[/imath] when [imath]C \geq 0[/imath], but for other values of [imath]C[/imath] I am not so sure (post #5).
Click to expand...
Do you mean that it is like we have a square root and we have to be careful to not get negative values inside it? You are not sure about that?

Isn't it sufficient to solve for this inequality?

[imath]Ce^{2x/5} +1 \geq 0[/imath]
 
mario99 said:
I see it now.


Do you mean that it is like we have a square root and we have to be careful to not get negative values inside it? You are not sure about that?

Isn't it sufficient to solve for this inequality?

[imath]Ce^{2x/5} +1 \geq 0[/imath]
Click to expand...
I was phrasing it as a hint ;)
 
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