LODEstruct is a data structure to represent an ordinary differential equation. It is created by Slode[DEdetermine].

The entries of an LODEstruct are a set of equations, representing the differential equation, and a set of function names, representing the dependent variables.

The data structure has an attribute table with the following entries:

L - the differential operator in diff notation

rhs - the right hand side of the equation

fun - the name of the dependent variable, for example $y$

var - the name of the independent variable, for example $x$

linear - $\mathrm{true}$ if L is a linear differential operator and $\mathrm{false}$ otherwise

ord - the order of L

coeffs - an Array of coefficients of L

polycfs - $\mathrm{true}$ if all coefficients are polynomial and $\mathrm{false}$ otherwise

d_max   - the maximum degree of polynomial coefficients

If the right hand side is a formal power series in the form $B\left(x\right)+\left(\sum _{n=N}^{\mathrm{\infty }}H\left(n\right)P_n\left(x\right)\right)$ where $B\left(x\right)$ is a polynomial in $x$, $P_n\left(x\right)$ is either ${\left(x-a\right)}^n$ or $\frac{1}{x^n}$, $a$ is the expansion point, and $H\left(n\right)$ is an expression in $n$, then it is represented as a RHSstruct data structure. The entries of an RHSstruct are the right hand side and the independent variable $x$. In addition, the data structure has an attribute table with following entries:

mvar - the name of the independent variable, $x$

index - the name of the summation index, $n$

point - the expansion point $a$, possibly $\mathrm{\infty }$

M - a nonnegative integer such that series coefficients are equal $H\left(n\right)$ for all $n\>M$; it satisfies $M=\max \left(N-1\,\mathrm{degree}\left(B\left(x\right)\,x\right)\right)$

initial - an Array of $M$ initial series coefficients

H - the expression $H\left(n\right)$

P_n - either ${\left(x-a\right)}^n$ or ${\left(\frac{1}{x}\right)}^n$

$\mathrm{with}\left(\mathrm{Slode}\right)\:$

$\mathrm{ode}≔\mathrm{diff}\left(y\left(x\right)\,x\right)\left(x-1\right)-y\left(x\right)=0$

$\mathrm{ode}≔\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\left(x-1\right)-y\left(x\right)=0$

$\mathrm{DEdetermine}\left(\mathrm{ode}\,y\left(x\right)\right)$

$\mathrm{LODEstruct}\left(\left\{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\left(x-1\right)-y\left(x\right)=0\right\}\,\left\{y\left(x\right)\right\}\right)$

$\mathrm{attributes}\left(\right)$

$table\left(\left[\mathrm{polycfs}=\mathrm{true}\,\mathrm{rhs}=0\,\mathrm{fun}=y\,\mathrm{linear}=\mathrm{true}\,\mathrm{coeffs}=\left[\mathrm{-1}\,x-1\,\text{⋯ 0 .. 1 Array}\right]\,\mathrm{d\_max}=1\,\mathrm{ord}=1\,\mathrm{var}=x\,L=\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\left(x-1\right)-y\left(x\right)\right]\right)$

$\mathrm{ode1}≔\mathrm{diff}\left(y\left(x\right)\,x\right)\left(x-1\right)-y\left(x\right)=x^3+2\left(\mathrm{Sum}\left(\frac{x^n}{n-3}\,n=4..\mathrm{\infty }\right)\right)$

$\mathrm{ode1}≔\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\left(x-1\right)-y\left(x\right)=x^3+2\left(\sum _{n=4}^{\mathrm{\infty }}\frac{x^n}{n-3}\right)$

$\mathrm{DEdetermine}\left(\mathrm{ode1}\,y\left(x\right)\right)$

$\mathrm{LODEstruct}\left(\left\{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\left(x-1\right)-y\left(x\right)=x^3+2\left(\sum _{n=4}^{\mathrm{\infty }}\frac{x^n}{n-3}\right)\right\}\,\left\{y\left(x\right)\right\}\right)$

$\mathrm{attributes}\left(\right)$

$table\left(\left[\mathrm{polycfs}=\mathrm{true}\,\mathrm{rhs}=\mathrm{RHSstruct}\left(x^3+2\left(\sum _{n=4}^{\mathrm{\infty }}\frac{x^n}{n-3}\right)\,x\right)\,\mathrm{fun}=y\,\mathrm{linear}=\mathrm{true}\,\mathrm{coeffs}=\left[\mathrm{-1}\,x-1\,\text{⋯ 0 .. 1 Array}\right]\,\mathrm{d\_max}=1\,\mathrm{ord}=1\,\mathrm{var}=x\,L=\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\left(x-1\right)-y\left(x\right)\right]\right)$

$\mathrm{attributes}\left(\left['\mathrm{rhs}'\right]\right)$

$table\left(\left[\mathrm{initial}=\left[0\,0\,0\,1\,\text{⋯ 0 .. 3 Array}\right]\,\mathrm{P\_n}=x^n\,M=3\,\mathrm{point}=0\,\mathrm{mvar}=x\,\mathrm{index}=n\,H=\frac{2}{n-3}\right]\right)$