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Q8E

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Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 259

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

Sturm–Liouville Form. A second-order equation is said to be in Sturm–Liouville form if it is expressed as $[p (t) y' (t)]' + q (t) y (t) = 0$ . Show that the substitutions $x_{1} = y, x_{2} = py'$ result in the normal form $x'_{1} = \frac{x_{2}}{p}, x_{2} = - {qx}_{1}$ . If $y (0) = a, y' (0) = b$ are the initial values for the Sturm–Liouville problem, what are $x_{1} (0) and x_{2} (0)$ ?

$x_{1} (0) = a x_{2} (0) = p (0) b$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Express the equation in form of x

Here given. $[p (t) y' (t)]' + q (t) y (t) = 0$

And

$x_{1} = y, x_{2} = py'$

The equation transforms as:

$x'_{2} + {qx}_{1} = 0 x'_{2} = - {qx}_{1} y' = \frac{x_{2}}{p} x'_{1} = \frac{x_{2}}{p}$

Step 2: The initial conditions.

The given initial conditions are $y (0) = a, y' (0) = b .$

Initial conditions after transformations;

$x_{1} (0) = a x_{2} (0) = p (0) y' (0) = p (0) b$

This is the required result.

Most popular questions for Math Textbooks

Use the result of Problem 31 to prove that all solutions to the equation\({\bf{y'' + }}{{\bf{y}}^{\bf{3}}}{\bf{ = 0}}\)remain bounded. [Hint: Argue that \(\frac{{{{\bf{y}}^{\bf{4}}}}}{{\bf{4}}}\) is bounded above by the constant appearing in Problem 31.]

In Problems 11–14, solve the related phase plane differential equation for the given system. Then sketch by hand several representative trajectories (with their flow arrows).

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and the solution curves for the related phase plane differential equation. Thereby proving that two trajectories lie on semicircles. What are the endpoints of the semicircles?

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