deletions | additions       diff --git a/develop_ode.tex b/develop_ode.tex  index f358302..5ab3525 100644  --- a/develop_ode.tex  +++ b/develop_ode.tex 
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\begin{equation}\label{eqn:psi}  \psi = \arctan{\frac{\omega}{z}} - \arctan{\frac{\rm{d}\omega}{\rm{d}z}}\ .  \end{equation}  Inserting equation~\ref{eqn:rho} and \ref{eqn:v0} into eqn.~\ref{eqn:4}, we arrive at an ordinary differential equation, equation (ODE),  that describes the functional form of the shape of the shock front: \begin{equation}  P(z) = \frac{3}{4}\rho_0v_0^2 = \frac{3}{4} \frac{\dot{M}}{4\pi v_{\infty}(z^2+\omega^2)} v_{\infty}^2 \sin^2(\psi)  \end{equation} 
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with  \begin{equation}  c = \frac{16\pi P(z)}{3 \dot{M} v_{\infty}}\ .  \end{equation} For general $P(z)$ this ODE needs to be solved numerically. The negative root describes the compression shock that heats up the gas; the positive root corresponds to an unphysical rarefraction shock.