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The rate at which a radioactive substance decays is proportional to the number of atoms, \(N(t)\), remaining at time \(t\). This is described by the first-order differential equation:$$\frac{dN}{dt} = -kN$$where \(k\) is the positive decay constant.
  1. Let the initial number of atoms be \(N_0\). State the initial condition for \(N(t)\).
  2. Verify that the general solution to this equation is \(N(t) = Ae^{-kt}\), where A is an arbitrary constant.
  3. Use the initial condition to find the particular solution for the number of atoms.

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