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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: Q1E (page 326)

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

Short Answer

Expert verified

Thus, the largest interval is $(- 鈭�, 0)$ .

Step by step solution

01

Solve the given equation

The given equation is $xy''' - 3 y' + e^{x} y = x^{2} - 1 .$

Divide both sides by x in the above equation,

$y''' - 3 (\frac{1}{x}) y' + e^{x} (\frac{1}{x}) y = x^{2} - 1 (\frac{1}{x})$

Compare with the standard form of a linear differential equation,

$y''' + p (x) y'' + q (x) y' + r (x) y = s (x)$

Therefore,

$q (x) = \frac{- 3}{x}, 鈥� 鈥� r (x) = \frac{e^{x}}{x}, 鈥� 鈥� s (x) = \frac{x^{2} - 1}{x}$

02

Step 2: Check the continuity

$q (x) = \frac{- 3}{x}$ is continuous whenever $x 鈮� 0 .$

$r (x) = \frac{e^{x}}{x}$ is continuous whenever $x 鈮� 0 .$ and

$s (x) = \frac{x^{2} - 1}{x}$ is continuous whenever $x 鈮� 0 .$

03

The largest interval (a, b)

Now overall q, r, and s are continuous in $鈭赌鈥� x 鈭� (- 鈭�, 鈥� 鈥� 0) 鈭� (0, 鈥� 鈥� 鈭�) .$

The initial condition is defined as $x_{0} = - 2$ and $- 2 鈭� (- 鈭�, 鈥� 鈥� 0) .$

Hence, the largest interval $(- 鈭�, 鈥� 鈥� 0) .$

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Most popular questions from this chapter

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.

$(x^{2} - 1) y''' + e^{x} y = \ln (x) y (\frac{3}{4}) = 1, 鈥� 鈥� y' (\frac{3}{4}) = y'' (\frac{3}{4}) = 0$

Find a general solution to the givenhomogeneous equation. ${(D + 1)}^{2} {(D 鈭� 6)}^{3} (D + 5) (D^{2} + 1) {(D^{2} + 4)}^{2} [y] = 0$

In Problems 1-6, use the method of variation of parameters to determine a particular solution to the given equation.

$y^{'''} + y^{'} = t a n x$

find a general solution to the given equation.

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

find a general solution to the given equation.

$y''' - 2 y'' - 5 y' + 6 y = e^{x} + x^{2}$

See all solutions

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