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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: Q21E (page 337)

use the annihilator method to determinethe form of a particular solution for the given equation. $u'' - 5 u' + 6 u = c o s 2 x + 1$

Short Answer

Expert verified

$u_{p} (x) = c_{3} + c_{4} \sin 2 x + c_{5} \cos 2 x$

Step by step solution

01

Solve the homogeneous of the given equation

The homogeneous of the given equation is

$(D^{2} - 5 D + 6) [u] = 0 鈬� (D - 2) (D - 3) [u] = 0$

The solution of the homogeneous is

$u_{h} (x) = c_{1} e^{2 x} + c_{2} e^{3 x}$ (1)

Let $g (x) = \cos 2 x$

Then

$(D^{2} + 2^{2}) [g] = 0 鈬� (D^{2} + 4) [g] = 0$

Let $h (x) = 1$

Then

$D [h] = 0$

Hence

$D (D^{2} + 4) [g + h] = 0$

Then, every solution to the given nonhomogeneous equation also satisfies

$D (D^{2} + 4) (D - 2) (D - 3) [u] = 0$

Then

$u (x) = c_{1} e^{2 x} + c_{2} e^{3 x} + c_{3} + c_{4} \sin 2 x + c_{5} \cos 2 x$ (2)

is the general solution to this homogeneous equation

We know $u (x) = u_{h} + u_{p}$

Comparing (1) & (2)

$u_{p} (x) = c_{3} + c_{4} \sin 2 x + c_{5} \cos 2 x$

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Most popular questions from this chapter

find a differential operator that annihilates the given function.

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

Use the annihilator method to show that if $a_{0} = 0$ and $a_{1} 鈮� 0$ in (4) andhas the form $f (x)$ given in (17), then equation (4) has a particular solution of the form $y_{p} (x) = x {B_{m} x^{m} + B_{m - 1} x^{m - 1} + 鈰� + B_{1} x + B_{0}}$

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

${\cos 2 x, 鈥� \cos^{2} x, 鈥� \sin^{2} x}$ on $(- 鈭�, 鈥� 鈭�)$

As an alternative proof that the functions $e^{r_{1} x}, e^{r_{2} x}, e^{r_{3} x}, ..., e^{r_{n} x}$ are linearly independent on (鈭�,-鈭�) when $r_{1}, r_{2}, . .. r_{n}$ are distinct, assume $C_{1} e^{r_{1} x} + C_{2} e^{r_{2} x} + C_{3} e^{r_{3} x} + . .. + C_{n} e^{r_{n} x}$ holds for all x in (鈭�,-鈭�) and proceed as follows:

(a) Because the ri鈥檚 are distinct we can (if necessary)relabel them so that $r_{1} > r_{2} > r_{3} > . .. > r_{n}$ .Divide equation (33) by to obtain $C_{1} + \frac{C_{2} e^{r_{2} x}}{e^{r_{2} x}} + \frac{C_{3} e^{r_{3} x}}{e^{r_{3} x}} + . .. + \frac{C_{n} e^{r_{n} x}}{e^{r_{n} x}} = 0$ Now let x鈫掆垶 on the left-hand side to obtainC1 = 0.(b) Since C1 = 0, equation (33) becomes

$C_{2} e^{r_{2} x} + C_{3} e^{r_{3} x} + . .. + C_{n} e^{r_{n} x}$ = 0for all x in(鈭�,-鈭�). Divide this equation by $e^{r_{2} x}$

and let x鈫掆垶 to conclude that C2 = 0.

(c) Continuing in the manner of (b), argue that all thecoefficients, C1, C2, . . . ,Cn are zero and hence $e^{r_{1} x}, e^{r_{2} x}, e^{r_{3} x}, . .., e^{r_{n} x}$ are linearly independent on(鈭�,-鈭�).

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.

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