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Q2E

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Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 326

Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

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.
$y''' - \sqrt{x} y = sinx y (π) = 0, y' (π) = 11, y'' (π) = 3$

Hence, the largest interval $[0, \infty) .$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Solve the given equation

The given equation is $y''' - \sqrt{x} y = sinx .$

Compare with the standard form of a linear differential equation,

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

Therefore,

$r (x) = - \sqrt{x}, s (x) = sinx$

Step 2:Check the continuity

$r (x) = - \sqrt{x}$ is continuous for all $x ⩾ 0$ .

$s (x) = sinx$ is continuous in $R$ .

Step 3:The largest interval (a, b)

Now overall r and s is continuous in $\forall x \in [0, \infty) .$

The initial condition is defined at $x_{0} = π .$

And $π \in [0, \infty) .$

Hence, the largest interval $[0, \infty) .$

Most popular questions for Math Textbooks

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

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

use the method of undetermined coefficients to determine the form of a particular solution for the given equation.

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

find a general solution to the given equation.

$y''' + 4 y'' + y' - 26 y = e^{- 3 x} s i n 2 x + x$

find a differential operator that annihilates the given function.

$x e^{3 x} c o s 5 x$

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.

$y''' - y'' + \sqrt{x - 1} y = tanx y (5) = y' (5) = y'' (5) = 1$

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