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anonymous
 4 years ago
Someone help me please :(
Consider the nonlinear 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.
anonymous
 4 years ago
Someone help me please :( Consider the nonlinear 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.

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TuringTest
 4 years ago
Best ResponseYou've already chosen the best response.0well a) is just about plugging in that guess for y into the DE and seeing if it's true

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0so i would just plug it into the LHS to check it?

TuringTest
 4 years ago
Best ResponseYou've already chosen the best response.0plug it into both sides:\[y=t\]\[\frac{dy}{dt}=\frac ty\]\[1=\frac t{t}=1\checkmark\]and that's that

TuringTest
 4 years ago
Best ResponseYou've already chosen the best response.0c) is the exact same thing, only you should be able to show that it's not the same on both sides

TuringTest
 4 years ago
Best ResponseYou've already chosen the best response.0part b) I think can be answered effectively by considering the domain of the equation

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0ok, so wait, for part c we sub 2t into both sides of the equation?

TuringTest
 4 years ago
Best ResponseYou've already chosen the best response.0\[y=2t\]\[\frac{dy}{dt}=2\]plug in those values into the DE and see if it's true

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0if your still there can you help me with part b by any chance!

TuringTest
 4 years ago
Best ResponseYou've already chosen the best response.0I'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

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0do 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?
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