Find an annihilator of x^2 + 2x^5

[magnesium12]

Homework Statement

Find an annihilator of x²+2x⁵.

Homework Equations

xⁿe^cx -----> [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]

Homework Statement

Find an annihilator of x²+2x⁵.

Homework Equations

xⁿe^cx -----> [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.

Doesn't the D⁶ operator also annihilate x²?

 

[magnesium12]

Doesn't the D⁶ operator also annihilate x²?

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?

 

[vela]

Homework Statement

Find an annihilator of x²+2x⁵.

Homework Equations

xⁿe^cx -----> [D-c]ⁿ⁺¹

[Q₁(D)Q₂(D)] = b₁(x) + b₂(x)

What does the last line mean? It's not clear to me.

The Attempt at a Solution

[/B]
x² -----> D³

2x⁵-----> D⁶

x² + 2x⁵ ------> (D³)(D⁶) = D¹⁸

The answer should be D⁶.

Why are you multiplying the two annihilators?

 

[Mark44]

(D³)(D⁶) = D¹⁸

Didn't notice this before -- is x³x⁶ = x¹⁸?

 

[Mark44]

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?

Dⁿ⁺¹ annihilates xⁿ and all lower powers of x. I don't see that this is related to the LCM in any way.

 

[vela]

Didn't notice this before -- is x³x⁶ = x¹⁸?

Heh...I didn't notice that either.