Nested alignments

Question

I want to make alignments that look like this:

which I have done by using

\begin{align}
         & \frac{dx}{dt} = y \qquad \frac{dy}{dt} = x \\
\implies & \frac{d^2x}{dt^2} = \frac{d}{dt}\bigg(\frac{dx}{dt} \bigg) \\
         & \; \; \; \; \; \; \, = \frac{dy}{dt} \\
         & \; \; \; \; \; \; \, = x \\
\implies & \frac{d^2x}{dt^2} = x \\
\implies & x(t) = Ae^t + Be^{-t}
\end{align}

Of course, it's kinda silly to do \; \; \; \; ... and align them manually. However, when I try putting a \begin{align} within another \begin{align} like this

\begin{align}
        \begin{align}

        \end{align}
\end{align}

it gives me an error.

What is the correct way of doing this?

---

EDIT:

\begin{alignat} still doesn't quite get me there:

\begin{alignat}{2}
         & \frac{dx}{dt} = y \qquad \frac{dy}{dt} = x \\
\implies & \frac{d^2x}{dt^2} && = \frac{d}{dt}\bigg(\frac{dx}{dt} \bigg) \\
         &                   && = \frac{dy}{dt} \\
         &                   && = x \\
\implies & \frac{d^2x}{dt^2} = x \\
\implies & x(t) = Ae^t + Be^{-t}
\end{alignat}

gives

Answer 1

You can use alignat for this:

\documentclass{article}

\usepackage{amsmath}

\begin{document}

\begin{alignat*}{2}
           &\quad& \frac{\mathrm{d}x}{\mathrm{d}t} &= y \qquad \frac{\mathrm{d}y}{\mathrm{d}t} = x \\
  \implies &&  \frac{\mathrm{d}^2x}{\mathrm{d}t^2} &= \frac{\mathrm{d}}{\mathrm{d}t} \biggl( \frac{\mathrm{d}x}{\mathrm{d}t} \biggr) \\
           &&                                      &= \frac{\mathrm{d}y}{\mathrm{d}t} \\
           &&                                      &= x \\
  \implies &&  \frac{\mathrm{d}^2x}{\mathrm{d}t^2} &= x \\
  \implies &&                                 x(t) &= Ae^t + Be^{-t}
\end{alignat*}

\end{document}

Answer 2

I wouldn't perform any alignment on the \implies symbols. To provide (visual) clarity of exposition, it suffices to perform alignment on the first instance of = in each line.

I can't help but remark that the 5th line seems to be redundant.

\documentclass{article} 
\usepackage{amsmath}
\begin{document} 
\begin{align}
\frac{dx}{dt}              &= y \qquad \frac{dy}{dt} = x \\
\implies \frac{d^2x}{dt^2} &= \frac{d}{dt}\biggl(\frac{dx}{dt}\biggr) \\
                           &= \frac{dy}{dt} \\
                           &= x \\
\implies \frac{d^2x}{dt^2} &= x \\
\implies x(t)              &= Ae^t + Be^{-t}
\end{align}
\end{document}

Answer 3

Nesting align environments directly doesn't work, because align environments start display math, so they can not be used inside math mode. Lucky for you, amsmath includes aligned as an alternative for cases like this (The t tells LaTeX to align the top line with the content outside of the environment):

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{align}
         & \frac{dx}{dt} = y \qquad \frac{dy}{dt} = x \\
\implies & \begin{aligned}[t]
             \frac{d^2x}{dt^2} &= \frac{d}{dt}\bigg(\frac{dx}{dt} \bigg) \\
                               &= \frac{dy}{dt} \\
                               &= x
           \end{aligned}\\
\implies & \frac{d^2x}{dt^2} = x \\
\implies & x(t) = Ae^t + Be^{-t}
\end{align}
\end{document}