the trick

Here is a pretty ubiquitous trick to solve differential equations of the second order differential equations. It is used to change a second order differential equation to a First-Order Differential Equations.

If you have a differential equation of the shape:

\begin{equation} x^{’’} = f(x,x’) \end{equation}

that, the second derivative is strictly a function between the first derivative value and the current value.

We are going to define a notation \(x’ = v\), which makes sense.

Here’s a general form:

\begin{equation} a\dv[2]{x}{t} + b \dv{x}{t} + cx = f(t) \end{equation}

see:

• solving homogeneous constant coefficient higher-order differential equations
• and more generally, using matrix exponentiation, solving homogeneous higher-order differential equations

solving homogeneous higher-order differential equations

This problem because easier if the right side is \(0\).

\begin{equation} a\dv[2]{x}{t} + b \dv{x}{t} + cx = 0 \end{equation}

The general goal to solve in this case is to make this a system of First-Order Differential Equations.

• generate
• prune a lot
• sample the values of conjectures based on the closest “unsolved” parts

There is a Self-Printing Turing Machine

Q

There’s a computable function \(q: \Sigma^{*} \to \Sigma ^{*}\) such that for every string \(w\), the computation \(q(w)\) produces the description of a Turing machine \(P_{w}\) such that on every input, it spits out \(w\) and then accepts.

B

For some turing machine code \(m\) making TM \(M\), the computable function \(B\) composes \(P_{m}\) from above and \(M\) (that is, the composition of \(P_{m}\) and \(M\) first disregards it input, prints \(m\), and give it to \(M\)).