Background

DiffResolvent.nb, here gives a method that solve quintic equations using differential equations.

There's a progress to transform polynomial equation into differential equation.

Then he gave an example:

$$x^5-5 x^3+5 x-t = 0$$

The roots of the equation satisfy the differential equation:

$$25 \left(4-t^2\right) x''(t)-25 t x'(t)+x(t)=0$$

Solve that and then can get the roots of quintic equation.

$$x(t)=c_1 \cos \left(\frac{1}{5} \arcsin\frac{t}{2}\right)-c_2 \sin \left(\frac{1}{5} \arcsin\frac{t}{2}\right)$$

We can easily check the results:

eq=x^5-5x^3+5 x-t
resolvent=25 (4-t^2) x''[t]-25 t x'[t]+x[t]

dso=DSolveValue[resolvent==0,x[t],t]/.Sqrt[4-t^2]->I Sqrt[t^2-4]
ser=Series[Evaluate[eq/.x->dso],{t,0,1},Assumptions->t\[Element]Reals];
sol=dso/.Solve[CoefficientList[ser,t]==0,{C[1],C[2]}];

SortBy[x/.NSolve[eq/.t->3,x],Im]
SortBy[sol/.t->3//N//Chop,Im]

I think this can't be solved by Solve.

Root[-3+5 #1-5 #1^3+#1^5&,1]//ToRadicals

Problem

But these codes were written in Mathematica 3.0 and can't run any more.

The behaviour of these functions differs from DifferentialRoot.

There's no intermediate process so I can't find whats wrong with these codes.

Goal

A function which convert the polynomial equation into differential equation.