Differential Equations

Consider the differential equation: dy/dt=y/t^2
a) Show that the constant function y1(t)=0 is a solution.
b)Show that there are infinitely many other functions that satisfy the differential equation, that agree with this solution when t<=0, but that are nonzero when t>0 [Hint: you need to define these functions using language like " y(t)=...when t<=0 and y(t)=...when t>0 and "]
c) Why doesn't this example contradict the Uniqueness Theorem?

I'm trying to do part b and after I separated and integrated I got
ln|y|=(-1/t)+C
I'm not sure if I can get C with the solution they gave in part a)y1(t)=0.
Anyways, I get y(t)=Ce^-(1/t). I don't know where to go from there.

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  1. dy/y = dt/t^2
    ln y = -1/t + c
    y = e^(-1/t+c) = e^c e^(-1/t)
    =C e^(-1/t) agree

    C can be anything so
    what if C = 0 ?
    then y(t) = 0 for all t

    if C = 0 for t</= 0 then C can be anything at all for t>0

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  2. If y(t) depends on what C is, then how this equation doesn't contradict the uniqueness theorem if it has many solutions?

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  3. Because the general solution contains an an arbitrary constant C. The value of C depends on your boundary conditions, for example if y =5 at t = 2
    then
    5 = C e^-(1/2)
    5 = C (.606)
    C = 8.24
    NOW your solution is unique.

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  4. scroll down through this:
    http://hyperphysics.phy-astr.gsu.edu/hbase/diff.html

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