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anonymous

  • 4 years ago

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.

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  1. TuringTest
    • 4 years ago
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    well a) is just about plugging in that guess for y into the DE and seeing if it's true

  2. anonymous
    • 4 years ago
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    so i would just plug it into the LHS to check it?

  3. TuringTest
    • 4 years ago
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    plug it into both sides:\[y=-t\]\[\frac{dy}{dt}=\frac ty\]\[-1=\frac t{-t}=-1\checkmark\]and that's that

  4. TuringTest
    • 4 years ago
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    c) is the exact same thing, only you should be able to show that it's not the same on both sides

  5. TuringTest
    • 4 years ago
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    part b) I think can be answered effectively by considering the domain of the equation

  6. anonymous
    • 4 years ago
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    ok, so wait, for part c we sub 2t into both sides of the equation?

  7. TuringTest
    • 4 years ago
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    \[y=2t\]\[\frac{dy}{dt}=2\]plug in those values into the DE and see if it's true

  8. TuringTest
    • 4 years ago
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    *yes, both sides

  9. anonymous
    • 4 years ago
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    ok thank you!

  10. TuringTest
    • 4 years ago
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    welcome

  11. anonymous
    • 4 years ago
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    if your still there can you help me with part b by any chance!

  12. TuringTest
    • 4 years ago
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    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

  13. anonymous
    • 4 years ago
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    ah ok thanks!!

  14. anonymous
    • 4 years ago
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    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?

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