Solving Differential Operator with D for 4(x^2)(D^2)+12xD+3

[asdf1]

for the following question:
[4(x^2)(D^2)+12xD+3)]y=0

my problem:
i only know how to solve this type of question if i change it back to its original form. however, if i want to solve it by just using D, how would i do that?

 

[quasar987]

What do you mean its original form?

And what are you expected to do with this equation anyway?

 

[asdf1]

i mean that if i changed the D to y``

 

[quasar987]

OH! Well I guess just treat Dy as dy/dx and solve as you would.

 

[saltydog]

OH! Well I guess just treat Dy as dy/dx and solve as you would.

I looked at it early this morning and even briefly posted what I though was the factored form of the differential operators and even though it resulted in a correct answer, my factored version was incorrect because the differential operators are NOT commutative. Turns out equations with functions of x are not easily factored for this reason, something I'm not clear on. So I would just treat it as a regular Euler-Cauchy equation and look for a solution of the form:

$$y=x^{m}$$

(unless someone can further comment about the factoring).

 

[asdf1]

i had the same problem as you when i tried to factor it too~

 

[Harmeet Singh]

This ia a quadratic in D.Just find out D using the formula used for quadratic equations and then proceed

 

[HallsofIvy]

This ia a quadratic in D.Just find out D using the formula used for quadratic equations and then proceed

The whole point is that this is an equation with VARIABLE coefficients. That method doesn't work with variable coefficients.

 

[asdf1]

@@ why doesn't it work for variable coefficients?