Question

Someone help me please :(
Consider the non-linear differential equation dy/dt = t/y . It was observed in Problem Set
2 that y = t is a solution to this differential equation.
(a) Verify that y = −t is also a solution.
(b) The solutions y = t and y = −t appear to cross at the point t = 0, y = 0. Why
does this not violate the uniqueness principle?
(c) By substituting into the differential equation verify that y = 2t is not a solution.

Answer 1

well a) is just about plugging in that guess for y into the DE and seeing if it's true

Answer 2

so i would just plug it into the LHS to check it?

Answer 3

plug it into both sides:\[y=-t\]\[\frac{dy}{dt}=\frac ty\]\[-1=\frac t{-t}=-1\checkmark\]and that's that

Answer 4

c) is the exact same thing, only you should be able to show that it's not the same on both sides

Answer 5

part b) I think can be answered effectively by considering the domain of the equation

Answer 6

ok, so wait, for part c we sub 2t into both sides of the equation?

Answer 7

\[y=2t\]\[\frac{dy}{dt}=2\]plug in those values into the DE and see if it's true

Answer 8

*yes, both sides

Answer 9

ok thank you!

Answer 10

welcome

Answer 11

if your still there can you help me with part b by any chance!

Answer 12

I'm not completely sure, but it doesn't really make sense to me as a way to violate the uniqueness theorem because the point x=0, y=0 is not in the domain of the differential equation

Answer 13

ah ok thanks!!

Answer 14

do you understand this by any chance?
Suppose that U^238 has a half life of 4.5 billion years, decaying (through a series of
relatively short lived intermediate atoms) to Pb^206. In a certain mineral sample there
are .31 times as many Pb^206 atoms as there are of U^238. If one assumes that the
mineral deposit contained no Pb^206 when it was formed and that no lead or uranium
have been added to or escaped from the sample (except through the natural decay
process) how old is the sample?