Factoring a differential equation into an operator product

Question

Following the fundamental theorem of algebra, I can (as stated here in Sec. 1.1.4) factor an $n$th-order linear ordinary differential equation $$ a_n \frac{d^n x}{dt^n} + a_{n-1} \frac{d^{n-1} x}{dt^{n-1}} +\cdots+a_1 \frac{dx}{dt}+a_0 = 0 $$ into $$ a_n \left( \frac{d}{dt} - r_1 \right) \left( \frac{d}{dt} - r_2 \right) \cdots \left( \frac{d}{dt} - r_n \right) x = 0 . $$

The Mathematica representation of the former is trivial, and I would represent the latter as

difOp[t, r] = (Dt[#, {t}] - r) &
an difOp[t, r1][ difOp[t, r2][x[t]] ] == 0

Is there any way to get the Mathematica kernel to connect the two, e.g. to compute the second from the first?

Answer 1

The differential operator in the first form can be written as

dd1[n_] := (Sum[a[k] D[#, {t, k}], {k, 0, n}]) &

and is applied for example as

dd1[1][x[t]]

a[0] x[t] + a[1] Derivative[1][x][t]

and

dd1[2] @x[t]

a[0] x[t] + a[1] Derivative[1][x][t] + a[2] (x^\[Prime]\[Prime])[t]

In the second (product) form we would perhaps try

d2[n_] := a[n] Expand[Product[D[#, t] - r[k], {k, 0, n}]] & 

But application leads to

d2[2][x[t]]

a[2] (-r[0] r[1] r[2] + r[0] r[1] Derivative[1][x][t] + 
   r[0] r[2] Derivative[1][x][t] + r[1] r[2] Derivative[1][x][t] - 
   r[0] Derivative[1][x][t]^2 - r[1] Derivative[1][x][t]^2 - 
   r[2] Derivative[1][x][t]^2 + Derivative[1][x][t]^3)

which is not the result we want.

So let us proceed step by step.

First we refrain from using D[] immediately but replace it by the symbol d:

d2[n_] := a[n] Expand[Product[d - r[k], {k, 0, n}]]

d2[1]

a[1] (d^2 - d r[0] - d r[1] + r[0] r[1])

Ok. Now in this expression we replace d^m by D[#,[t,m}] for m=1, 2, ..., n which gives us

dd2[n_] := a[n]*Expand[Product[d - r[k], {k, 1, n}]] /. 
         d^(m_) -> D[#1, {t, m}] /. d -> D[#1, {t, 1}] & 

Notice that we need to replace the the first power of d separately because it does not match te pattern d^m_.

Applying it

dd2[2][x[t]]

a[2] (r[1] r[2] - r[1] Derivative[1][x][t] - 
   r[2] Derivative[1][x][t] + (x^\[Prime]\[Prime])[t])

shows that it is correct.

Now the quantities r[i] are of course the roots of the equation

eq[n_] := 0 == Sum[a[k] r^k, {k, 0, n}]

eq[1]

0 == a[0] + r a[1]

eq[2]

0 == a[0] + r a[1] + r^2 a[2]

These can be written explicitly thus

r[n_, k_] := Root[Sum[a[j] #1^j , {j, 0, n}] &, k]

r[5, 2]

Root[a[0] + a[1] #1 + a[2] #1^2 + a[3] #1^3 + a[4] #1^4 + a[5] #1^5 &, 2]

Summarizing, we have provided all contructs requested in the question.

Regards, Wolfgang