Q1E Let A=D-1,B=D+2,C=D2+D-2, where ... [FREE SOLUTION]

91影视

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 5: Q1E (page 249)

Let $A = D - 1, B = D + 2, C = D^{2} + D - 2,$ where $D = \frac{d}{dt}$ . For $y = t^{3} - 8$ , compute
(a) $A [y]$
(b) $B [A [y]]$
(c) $B [y]$
(d) $A [B [y]]$
(e) $C [y]$

Short Answer

Expert verified

(a) The solution of $A [y]$ is $- t^{3} + 3 t^{2} + 8$ .

(b) The solution of $B [A [y]]$ is $- 2 t^{3} + 3 t^{2} + 6 t + 16$

(d) The solution of $A [B [y]]$ is $- 2 t^{3} + 3 t^{2} + 6 t + 16$

(e) The solution of $C [y]$ is $- 2 t^{3} + 3 t^{2} + 6 t + 16$

Step by step solution

General form

Elimination Procedure for 2 脳 2 Systems:

To find a general solution for the system

$L_{1} [x] + L_{2} [y] = f_{1}, L_{3} [x] + L_{4} [y] = f_{2},$

a. Make sure that the system is written in operator form.

b. Eliminate one of the variables, say, y, and solve the resulting equation for x(t).

If the system is degenerate, stop! A separate analysis is required to determine whether or not there are solutions.

c. (Shortcut) If possible, use the system to derive an equation that involves y(t) but not its derivatives. [Otherwise, go to step (d).] Substitute the found expression for x(t) into this equation to get a formula for y(t). The expressions for x(t), y(t) gives the desired general solution.

d. Eliminate x from the system and solve for y(t). [Solving for y(t) gives more constants----in fact, twice as many as needed.]

e. Remove the extra constants by substituting the expressions for x(t) and y(t) into one or both of the equations in the system. Write the expressions for x(t) and y(t) in terms of the remaining constants.

Evaluate the given equation

Given that, $A = D - 1, B = D + 2, C = D^{2} + D - 2,$ where $D = \frac{d}{dt}$ .

And $y = t^{3} - 8$ .

Then compute the given parts,

$A [y] = (D - 1) [t^{3} - 8] = \frac{d}{dt} [t^{3} - 8] - [t^{3} - 8] = 3 t^{2} - t^{3} + 8 = - t^{3} + 3 t^{2} + 8$

Compute the parts (b) and (c)

$B [A [y]] = (D + 2) [- t^{3} + 3 t^{2} + 8] = \frac{d}{dt} [- t^{3} + 3 t^{2} + 8] + 2 [- t^{3} + 3 t^{2} + 8] = - 3 t^{2} + 6 t - 2 t^{3} + 6 t^{2} + 16 = - 2 t^{3} + 3 t^{2} + 6 t + 16$

$B [y] = (D + 2) [t^{3} - 8] = \frac{d}{dt} [t^{3} - 8] + 2 [t^{3} - 8] = 3 t^{2} + 2 t^{3} - 16$

Compute the parts (d) and (e)

$A [B [y]] = (D - 1) [2 t^{3} + 3 t^{2} - 16] = \frac{d}{dt} [2 t^{3} + 3 t^{2} - 16] - [2 t^{3} + 3 t^{2} - 16] = 6 t^{2} + 6 t - 2 t^{3} - 3 t^{2} + 16 = - 2 t^{3} + 3 t^{2} + 6 t + 16$

$C [y] = (D^{2} + D - 2) [t^{3} - 8] = \frac{d^{2}}{{dt}^{2}} [t^{3} - 8] + \frac{d}{dt} [t^{3} - 8] - 2 [t^{3} - 8] = 6 t + 3 t^{2} - 2 t^{3} + 16$

So, the solutions are founded.

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91影视

Let $A = D - 1, B = D + 2, C = D^{2} + D - 2,$ where $D = \frac{d}{dt}$ . For $y = t^{3} - 8$ , compute
(a) $A [y]$
(b) $B [A [y]]$
(c) $B [y]$
(d) $A [B [y]]$
(e) $C [y]$

Short Answer

Step by step solution

General form

Evaluate the given equation

Compute the parts (b) and (c)

Compute the parts (d) and (e)

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91影视

Let A=D-1,B=D+2,C=D2+D-2, where D=ddt. For y=t3-8, compute(a)Ay(b)BAy(c)By(d)ABy(e)Cy

Short Answer

Step by step solution

General form

Evaluate the given equation

Compute the parts (b) and (c)

Compute the parts (d) and (e)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Calculus

Pure Maths

Mechanics Maths

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Let $A = D - 1, B = D + 2, C = D^{2} + D - 2,$ where $D = \frac{d}{dt}$ . For $y = t^{3} - 8$ , compute
(a) $A [y]$
(b) $B [A [y]]$
(c) $B [y]$
(d) $A [B [y]]$
(e) $C [y]$