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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 a strictly positive decay constant. Let \(N_0\) be the number of atoms at \(t=0\).
  1. Determine the general solution of this differential equation on \([0, +\infty)\).
  2. Use the initial condition \(N(0) = N_0\) to find the unique solution for \(N(t)\).

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