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\usepackage{amsmath}

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Here are some examples of various equations and equation environments in \LaTeX. 
Look them over and see how they work.

Here are some basic examples:
\begin{equation}
	a_0 = \frac{F}{m} = \frac{P_0 A}{m}
\end{equation}
Note the difference between equation and equation*:
\begin{equation*}
	P_0 A t = m\left(1+\frac{\rho x A}{m}\right)v
\end{equation*}

Equation array (eqnarray) lets you align equations.
Here are some examples of eqnarray, numbered and unnumbered.

\begin{eqnarray*}
	P_0(x)	&=& 1\\
	P_1(x)	&=& x\\
	P_2(x)	&=& \frac{1}{2}(3x^2-1)\\
	P_3(x)	&=& \frac{1}{2}(5x^3-3x)\\
	P_4(x)	&=& \frac{1}{8}(35x^4-30x^2+3)\\
	\vdots\\
	P_l(x)	&=& \frac{1}{2^l l!}\left(\frac{d}{dx}\right)^l(x^2-1)^l
\end{eqnarray*}

\begin{eqnarray}
	V	&=& \frac{2\pi R^3}{3}\int_0^\pi [1 + \delta]^3 P_0^3\sin\theta\,d\theta\\
		&\approx& \frac{2\pi R^3}{3}[1+3\delta] \int_0^\pi P_0 P_0\sin\theta\,d\theta\\
		&=& \frac{4\pi R^3}{3}[1+3\delta]
\end{eqnarray}

\verb:\[ : and \verb:\]: are shortcuts for \verb:\begin{equation*}: and \verb:\end{equation*}:.
\[	x(t) = \lambda\left[\sqrt{1+\frac{a_0t^2}{\lambda}} - 1\right] \]

\verb:$$: does the same thing, 
$$ v(t) = v_{\max}\left(1+\frac{\lambda}{a_0t^2}\right)^{-1/2} $$
but \$ creates an inline equation such as $v=\frac{\partial x}{\partial t}$.

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