If Y1(t) and Y2(t) are solutions of ay''+by'+cy (where a,b, and c are positive constants), show that Y1(t)-Y2(t) -0 as t-infinity. Is this result true if b=0?

Question

If Y1(t) and Y2(t) are solutions of ay''+by'+cy (where a,b, and c are positive constants), show that Y1(t)-Y2(t) ->0 as t->infinity. Is this result true if b=0?

anonymous · Answer

not sure how much knowledge you have of DE's. and which variables you are familiar with.

the solution is c1e^(root#1*t) + c2e^(root#2*t)

and the polynomial factored out, for b > 0, will look like this:

[let lambda be the characteristic equation variable]

$$( \lambda + R_1)(\lambda + R_2)  = 0$$so the solution will be:
$$y(t) = y_1(t) + y_2(t) = c_1e^{-R_1} + c_2e^{-R_2}$$

as t approaches infinity, this equals zero.


this is NOT NECESSARILY TRUE for b = 0

anonymous · Answer

dont hesitate tp ask questions for clarification because i have no idea what your current knowledge is

anonymous · Answer

Yeah, I'm taking intro to DE in college right now, and this is a homework problem haha. I think I get the first part. Can you explain why it doesn't work for b=0 though?

anonymous · Answer

OH WAIT, I get it. Because it would factor out to (lambda + R1)(lambda - R2) if b was 0?

anonymous · Answer

yes. one solution will go to inifity as t -> infinity

anonymous · Answer

Ok that totally makes sense now. Thanks so much!!

anonymous · Answer

glad i could help ^_^

anonymous · Answer

Oh, actually, one more question if you don't mind... what if the polynomial has imaginary solutions?

anonymous · Answer

Nevermind, I figured it out :)