How to solving the ODE $\frac{dp}{dt}=-\frac{\lambda}{p}$?

Question

$p=p(t)$ is a function about $t$. I try to do so: $$ \frac{dp}{dt}=-\frac{\lambda}{p}~\Rightarrow ~p dp=-\lambda dt~ \Rightarrow~\int p dp=\int -\lambda dt ~\Rightarrow~ \frac{1}{2}p^2=-\lambda t ~\Rightarrow~ p^2=-2\lambda t $$

But the answer of book is $p^2=1-2\lambda t$ ,I don't know why there is 1. The question is from the change of Einstein manifold under Ricci flow.

Answer 1

The general solution of the ODE has an integration constant : $p^2=-2\lambda t +c$ and the value $c=1$ correspond to the inital condition $p^2=1$.