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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: Q17E (page 337)

find a differential operator that annihilates the given function.
$x^{2} e^{- x} s i n 2 x$

Short Answer

Expert verified

${(D^{2} + 2 D + 5)}^{3}$ is the differential operator that annihilates the given function.

Step by step solution

01

Any nonhomogeneous term of the form f(x)=xkeαxcosβx OR xkeαxsinβx satisfies (D-α)2+β2m[f]=0for K=0,1,...,m-1

Let the function be $f (x) = x^{2} e^{- x} \sin 2 x$

Hence ${[{(D + 1)}^{2} + 2^{2}]}^{3} [f] = 0$

Then ${[{(D + 1)}^{2} + 2^{2}]}^{3} = {(D^{2} + 2 D + 1 + 4)}^{3} = {(D^{2} + 2 D + 5)}^{3}$ is the differential operator that annihilates the given function.

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Most popular questions from this chapter

. find a differential operator that annihilates the given function.

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

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 (∞,-∞) 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 (∞,-∞) and proceed as follows:

(a) Because the ri’s 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→∞ 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(∞,-∞). Divide this equation by $e^{r_{2} x}$

and let x→∞ 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(∞,-∞).

find a general solution to the given equation.

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

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

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

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

Determine the largest interval (a, b) for which Theorem 1 guarantees the existence of a unique solution on (a, b) to the given initial value problem.

$xy''' - 3 y' + e^{x} y = x^{2} - 1 y (- 2) = 1, y' (- 2) = 0, y'' (- 2) = 2$

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