I am reading a nice book (Quarks and Leptons, by Halzen and Martin) about particle physics. It states that the general form of the propagator of a virtual particle is:
\dfrac{i\sum_{\text{spins}}}{p^2 - m^2}
I see that this is the case for the Dirac propagator...
I assumed you were familiar with the definition of first integral. A first integral for the system y' = g(y), g : D \subseteq \mathbb{R}^n \rightarrow \mathbb{R}^n is a C^1 scalar function E : D \rightarrow \mathbb{R} constant on every solution of the system. In other words, if \phi : I...
Your equation is equivalent to a first-order system, if we let F = u and F' = v:
\begin{cases}\dot{u} = v \\ \dot{v} = ku-v^2\end{cases}
From there you can quite easily find a first integral and then separate variables in the second equation:
\begin{cases}\dot{u} = v \\ \dot{v}^2 = v^4 +...