Understanding Differential Operators and their Relationship to O.D.E. Equations

[asdf1]

why is the answer to a differential operator the same as the answer to the original O.D.E. equation?

 

[HallsofIvy]

That question doesn't even makes sense. A "differential operator" is neither an equation not a "problem" and doesn't have an answer!

Could you please rephrase your question.

 

[dextercioby]

I think he means the "fundamental solution" of the differential operator.

Daniel.

 

[asdf1]

for example:
(D^2-d-2)y=0
if you solve D, which is D=2,-1
it's the same as solving the equation~

 

[dextercioby]

In the light of your latest discoveries, your question still doesn't make too much sense.

Daniel.

 

[HallsofIvy]

I think he means the "fundamental solution" of the differential operator.
Daniel.

Actually, I would argue that that doesn't make sense either- an operator is not an equation. An equation may have a solution, but not the operator!

for example:
(D^2-d-2)y=0
if you solve D, which is D=2,-1

Okay, I can understand that, although your terminology is still odd!
I presume you are referring to the differential equation:
(D²- D- 2)y= 0 where "D" is the differential operator d/dx (or d/dt). D²- D- 2 would also be a linear differential operator.

However, solving the equation D²- D- 2= 0 to get D= 2 or D= -1 is what mathematicians call "abuse of terminology". If you intend D to be "d/dx", it clearly doesn't make sense to turn around and say that D= 2!
It is, though, convenient shorthand and we do it all the time. It is convenient shorthand for the "characteristic equation". If we were to look for a solution of the form y= e^{\lambda x}, putting that into the equation would give \lambda^2 e^{\lambda x}- \lambda e^{\lambda x}- 2e^{\lambda x}= 0 or \left(\lambda^2- \lambda- 2\right)e^{\lambda x}= 0. Since y= e^{\lambda x} is never 0, we must have \lambda^2- \lambda- 2= 0, the characteristic equation.
That "looks like" the original equation, especially in operator notation (which is the main reason for using it) because of that very nice property of exponentials:
\frac{d^n(e^{\lambda x}}{dx^n}= \lambda^n e^{\lambda x}

Caution! This is only true for a very limited (though important) class of differential equations: linear equations with constant coefficients.

 

[asdf1]

sorry, i didn't make myself clear~
for the example above, usually the normal way is to suppose that
y=ce^(namda)x
and find namda
but the answer to namda is the same as finding the answer to the differential operator~

 

[HallsofIvy]

Yes, that was what I just said. Oh, by the way, the Greek letter is "lambda", not "nambda".