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What integration strategy is used on the second-order differential equation on this page? Specifically, when solving the following equation,

$$ \frac{d^{2}w}{dz^{2}}-zw(z) = 0 $$

how do you know to set $\mathbf{y}=\left[\frac{dw}{dz},w\right]$ and $t=z$ and get to the following?

$$ \frac{d\mathbf{y}}{dt}=\left[\begin{array}{c} ty_{1}\\ y_{0}\end{array}\right]=\left[\begin{array}{cc} 0 & t\\ 1 & 0\end{array}\right]\left[\begin{array}{c} y_{0}\\ y_{1}\end{array}\right]=\left[\begin{array}{cc} 0 & t\\ 1 & 0\end{array}\right]\mathbf{y}. $$

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up vote 4 down vote accepted

To replace $z$ by $t$ is arbitrary, the point is that many authors treat linear ODE with the picture in mind of a system evolving in time, therefore the variable is mostly denoted by "$t$" for "time".

Another point is that it is a convention to reduce linear higher order differential equations $$ a_n (t) w^{(n)} + ... + a_1(t) w =0 $$ to a linear system of differential equations of order one by introducing the "auxiliary" functions $$ w_0 (t) := w(t) $$ $$ w_1 (t) := w^{(1)}(t) $$ etc. so that every linear differential equation can be written in the form $$ \frac{d}{d t} \vec{w} = A(t) \vec{w} $$ with an appropriate matrix $A(t)$ and the vector $\vec{w}(t) := (w_0(t), ..., w_n(t))$. Such a reformulation is always possible, it is equivalent to the original one. You treat the lower order of differentials (1 instead of $n$) for the number of equations ($n$ instead of 1). This formulation is the "standard formulation" for linear ordinary differential equations: It is useful because one can deduce information about the system from information about the matrix $A$, i.e. by using linear algebra, for this reason many authors use mainly this reformulation.

For example, if $A$ is diagonalizable and independent of $t$ (then one talks about an "autonomous" system of ODE), by determining a basis of eigenvectors and formulating the equation with respect to this basis, you immediately get a solution of the equation.

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