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Fundamentals Of Differential Equations And Boundary Value Problems

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

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 6: Q24E (page 332)

Let $P (r_{1}) = a_{n} r_{1}^{n} + a_{n 鈭� 1} r_{1}^{n 鈭� 1} + . .. + a_{1} r_{1} + a_{0}$ be a polynomialwith real coefficients $a_{n}, . ..., a_{0}$ . Prove that if r₁ is azero of $P (r)$ , then so is its complex conjugate r₁. [Hint:Show that $\overset{炉}{P (r)} = P (\overset{炉}{r})$ , where the bar denotes complexconjugation.]

Short Answer

Expert verified

If r₁ is a zero of the given polynomial P(r) then so is its complex conjugate r₁.

Step by step solution

01

Step 1: Complex Conjugation

A complex conjugate of a complex number is another complex number that has the same real part as the original complex number and the imaginary part has the same magnitude but opposite sign. The product of a complex number and its complex conjugate is a real number.

02

Use of Complex Conjugation properties

Suppose complex number r₁ is a zero of the given polynomial P(r). Hence P(r₁)=0:

$P (r_{1}) = 0 鈬� \overset{炉}{P (r_{1})} = \overset{炉}{0} = 0$

Using properties of complex conjugation:

$\begin{array}{l} \overset{炉}{P (r_{1})} = \overset{炉}{(a_{n} r_{1}^{n} + a_{n 鈭� 1} r_{1}^{n 鈭� 1} + ... + a_{1} r_{1} + a_{0})} \\ = a_{n} \overset{炉}{r_{1}^{n}} + a_{n 鈭� 1} \overset{炉}{r_{1}^{n 鈭� 1}} + ... + a_{1} \overset{炉}{r_{1}} + a_{0} \\ = P (\overset{炉}{r_{1}}) = 0 \end{array}$

Hence, the final answer is:

Ifr₁ is a zero of the given polynomial P(r) then so is its complex conjugate r₁.

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Most popular questions from this chapter

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.

${e^{3 x}, 鈥� e^{5 x}, 鈥� e^{- x}}$ on $(- 鈭�, 鈥� 鈭�)$

Use the annihilator method to show that if $f (x)$ in (4) has the form $f (x) = a c o s 尾 x + b s i n 尾 x,$

then equation (4) has a particular solution of the form

(18) $y_{p} (x) = x^{s} {A c o s 尾 x + B s i n 尾 x}$ ,where sis chosen to be the smallest nonnegative integer such that $x^{3} c o s 尾 x$ and $x^{3} \sin 尾 x$ are not solutions to the corresponding homogeneous equation

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

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

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

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

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.

find a differential operator that annihilates the given function.

$x^{2} - e^{x}$

use the annihilator method to determinethe form of a particular solution for the given equation. $y'' - 6 y' + 10 y = e^{3 x} - x$

See all solutions

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