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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

Deflection of a Beam Under Axial Force. A uniform beam under a load and subject to a constant axial force is governed by the differential equation

$y^{(4)} (x) - k^{2} y^{''} (x) = q (x), 0 < x < L,$

where is the deflection of the beam, L is the length of the beam, k²is proportional to the axial force, and q(x) is proportional to the load (see Figure 6.2).

(a) Show that a general solution can be written in the form

$y (x) = C_{1} + C_{2} x + C_{3} e^{k x} + C_{4} e^{- k x} + \frac{1}{k^{2}} \int q (x) x d x - \frac{x}{k^{2}} \int q (x) d x + \frac{e^{k x}}{2 k^{3}} \int q (x) e^{- k x} d x - \frac{e^{- k x}}{2 k^{3}} \int q (x) e^{k x} d x$

(b) Show that the general solution in part (a) can be rewritten in the form

$y (x) = c_{1} + c_{2} x + c_{3} e^{k x} + c_{4} e^{- k x} + \int_{0}^{x} q (s) G (s, x) d s,$

where

$G (s, x) : = \frac{s - x}{k^{2}} - \frac{s i n h [k (s - x)]}{k^{3}} .$

(c) Let q(x)=1 First compute the general solution using the formula in part (a) and then using the formula in part (b). Compare these two general solutions with the general solution

$y (x) = B_{1} + B_{2} x + B_{3} e^{k x} + B_{4} e^{- k x} - \frac{1}{2 k^{2}} x^{2},$

which one would obtain using the method of undetermined coefficients.

use the method of undetermined coefficients to determine the form of a particular solution for the given equation.

$y''' + 3 y'' - 4 y = e^{- 2 x}$

. find a differential operator that annihilates the given function.

$x^{4} - x^{2} + 11$

Find a general solution for the given

linear system using the elimination method of Section 5.2.

$\begin{array}{l} \frac{d^{3} x}{d t^{3}} - x + \frac{d y}{d t} + y = 0 \\ \frac{d x}{d t} - x + y = 0 \end{array}$

Determine the intervals for which Theorem guarantees the existence of a solution in that

(a) $y^{(4)} - (l n x) y^{''} + x y^{'} + 2 y = c o s 3 x$

(b) $(x^{2} - 1) y^{'''} + (s i n x) y^{''} + \sqrt{x + 4} y^{'} + e^{x} y = x^{2} + 3$

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

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.3x2-6x+1

Short Answer

Step by step solution

Differentiate the function continuously

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Theoretical and Mathematical Physics

Geometry

Decision Maths

Applied Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

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