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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: Q Review Problems-1E (page 343)

Determine the intervals for which Theorem guarantees the existence of a solution in that
(a) $y^{(4)} - (l n x) y^{''} + x y^{'} + 2 y = c o s 3 x$
(b) $(x^{2} - 1) y^{'''} + (s i n x) y^{''} + \sqrt{x + 4} y^{'} + e^{x} y = x^{2} + 3$

Short Answer

Expert verified

$(0, 鈭�)$ is interval in which theorem guarantee unique solution.
$(- 4, - 1), (- 1, 1)$ and $(1, 鈭�)$ are interval in which theorem guarantee unique solution.

Step by step solution

01

Determine the interval for which theorem guarantee unique solution.

$y^{4} - \ln x y^{''} + x y^{'} + 2 y = \cos 3 x$

Here 1, $\ln x, x, 2$ and $\cos 3 x$ have to be continuous.

Hence,

$(0, 鈭�)$ is interval in which theorem guarantee unique solution.

02

Determine the interval for which theorem guarantee unique solution

$(x^{2} - 1) y^{'''} + \sin x y^{''} + \sqrt{x + 4} y^{'} + e^{x} y = x^{2} + 3 y^{'''} + \frac{\sin x}{(x^{2} - 1)} y^{''} + \frac{\sqrt{x + 4}}{(x^{2} - 1)} y^{'} + \frac{e^{x}}{(x^{2} - 1)} y = \frac{x^{2} + 3}{(x^{2} - 1)}$

All quotient functions have to be continuous.

$x^{2} - 1 鈮� 0 and \sqrt{x + 4} > 0 x 鈭� (- 4, - 1) 鈭� (- 1, 1) 鈭� (1, 鈭�)$

Hence,

$(- 4, - 1), (- 1, 1)$ and $(1, 鈭�)$ are interval in which theorem guarantee unique solution.

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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.

$x \sqrt{x + 1} y''' - y' + xy = 0 y (\frac{1}{2}) = y' (\frac{1}{2}) = - 1, 鈥� 鈥� y'' (\frac{1}{2}) = 1$

In Problems 1-6, use the method of variation of parameters to determine a particular solution to the given equation. $y^{'''} + y^{'} = s e c 胃 t a n 胃, 0 < 胃 < 蟺 / 2$

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

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

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

Find a general solution for the differential equation with x as the independent variable:

$y''' + 5 y'' + 3 y' 鈭� 9 y = 0$

See all solutions

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