The initial value problem (IVP) for the ode

\[ \frac{d Y}{d t}=-C Y^{2} \]

with \(C\) being a positive constant and initial condition \(Y(0)=Y_{0} \geq 0\) generates a mapping \(Y(t): Y(0) \rightarrow R^{1}\).

12.1.1 Compute the solution of the IVP.

12.1.2 Let the initial value \(Y_{0}\) be a random variable with \(\operatorname{Pdf} f_{Y}(y ; 0)\), determine the Pdf of \(Y(t)\) for \(t \geq 0\).

12.1.3 Determine the asymptotic Pdf \(f_{Y}(y ; \infty)\).