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Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset Vaia Explanations$ Math

9th Edition · 616 pages

ISBN: 9780321977069

Chapter 6: Q12E (page 337)

find a differential operator that annihilates the given function.
$3 x^{2} - 6 x + 1$

Short Answer

Expert verified

$D^{3}$ is the differential operator that annihilates the given function.

Step by step solution

Differentiate the function continuously

Let the function be $f (x) = 3 x^{2} - 6 x + 1$

Then

$f' (x) = 6 x - 6 f'' (x) = 6 f''' (x) = 0$

Hence $D^{3} [f] = 0$

Then $D^{3}$ is the differential operator that annihilates the given function.

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Most popular questions from this chapter

Find a general solution to the Cauchy-Euler equation $x^{3} y^{'''} - 3 x^{2} y^{''} + 6 x y^{'} - 6 y = x^{- 1}, x > 0,$

given that ${x, x^{2}, x^{3}}$ is a fundamental solution set for the corresponding homogeneous equation

Find a general solution to the Cauchy-Euler equation $x^{3} y^{'''} - 3 x y^{'} + 3 y = x^{4} c o s x, x > 0$

Given that ${e^{x}, e^{- x}, e^{2 x}}$ is a fundamental solution set for the homogeneous equation corresponding to the equation $y^{'''} - 2 y^{''} - y^{'} + 2 y = g (x),$

determine a formula involving integrals for a particular solution.

In Problems 1-6, use the method of variation of parameters to determine a particular solution to the given equation.

$y^{'''} - 2 y^{''} + y^{'} = x$

Find a general solution to $y^{(4)} + 2 y^{(3)} + 4 y'' + 3 y' + 2 y = 0$ by using Newton’s method to approximate numerically the roots of the auxiliary equation. [Hint: To find complex roots, use the Newton recursion formula $z_{n + 1} = z_{n} - \frac{f (z_{n})}{f' (z_{n})}$ and start with a complex initial guess z0.]

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find a differential operator that annihilates the given function.3x2-6x+1

Short Answer

Step by step solution

Differentiate the function continuously

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Most popular questions from this chapter

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find a differential operator that annihilates the given function.
$3 x^{2} - 6 x + 1$