Doesn't the D6 operator also annihilate x2?magnesium12 said:Homework Statement
Find an annihilator of x2+2x5.
Homework Equations
xnecx -----> [D-c]n+1
[Q1(D)Q2(D)] = b1(x) + b2(x)
The Attempt at a Solution
[/B]
x2 -----> D3
2x5-----> D6
x2 + 2x5 ------> (D3)(D6) = D18
The answer should be D6.
Mark44 said:Doesn't the D6 operator also annihilate x2?
What does the last line mean? It's not clear to me.magnesium12 said:Homework Statement
Find an annihilator of x2+2x5.
Homework Equations
xnecx -----> [D-c]n+1
[Q1(D)Q2(D)] = b1(x) + b2(x)
Why are you multiplying the two annihilators?The Attempt at a Solution
[/B]
x2 -----> D3
2x5-----> D6
x2 + 2x5 ------> (D3)(D6) = D18
The answer should be D6.
Didn't notice this before -- is x3x6 = x18?magnesium12 said:(D3)(D6) = D18
Dn+1 annihilates xn and all lower powers of x. I don't see that this is related to the LCM in any way.magnesium12 said:I guess it does. So does that mean there can be multiple annihilators and they'll all work? Or should you always pick the least common multiple?
Heh...I didn't notice that either.Mark44 said:Didn't notice this before -- is x3x6 = x18?
An annihilator is an algebraic expression that, when multiplied with a given polynomial, results in a zero polynomial. In other words, it "annihilates" the given polynomial.
To find an annihilator of a polynomial, you must first determine the highest power of the variable in the polynomial. Then, you can construct an annihilator using the form (x-a)^n, where a is a constant and n is the highest power of the variable in the polynomial.
First, we determine the highest power of the variable, which in this case is 5. Then, we construct the annihilator using the form (x-a)^n. In this case, a=0 and n=5, so the annihilator is (x-0)^5, or simply x^5.
An annihilator is important in mathematics because it allows us to solve differential equations and perform other complex mathematical operations. It also has applications in engineering, physics, and other scientific fields.
Yes, there are other methods for finding an annihilator, such as using the method of undetermined coefficients or using the Laplace transform. However, the method of constructing an annihilator using the form (x-a)^n is the most commonly used and straightforward method.