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

The solution to the initial value problem
$\begin{array}{l} x y^{''} (x) + 2 y^{'} (x) + x y (x) = 0 \\ y (0) = 1, y^{'} (0) = 0 \end{array}$
has derivatives of all orders at $x = 0$ (although this is far from obvious). Use L'Hôpital's rule to compute the Taylor polynomial of degree 2 approximating this solution.

The required polynomial is, $p_{3} (x) = 1 - \frac{x^{2}}{6}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1:To Find the Taylor polynomial of degree

The formula for the Taylor polynomial of degree n centered at , $x_{0}$ approximating a function $f (x)$ possessing n derivatives at , $x_{0}$ is given by

$p_{n} (x) = f (x_{0}) + f^{'} (x_{0}) \times (x - x_{0}) + f^{''} (x_{0}) \times \frac{{(x - x_{0})}^{2}}{2!} + \dots + f^{n} (x_{0}) \times \frac{{(x - x_{0})}^{n}}{n!}$

The differential equation is given as

$x y^{''} (x) + 2 y^{'} (x) + x y (x) = 0$

It is given that for the function , $y (x)$

$y (0) = 1 and y^{'} (0) = 0$

For applying the L’Hospital rule we need to see that our differential equation satisfies $\frac{0}{0}$ form,

$\begin{matrix} x y^{''} (x) + 2 y^{'} (x) + x y (x) = 0 \\ y^{''} (x) + \frac{2 y^{'} (x)}{x} + y (x) = 0 \\ y^{''} (x) = - y (x) - \frac{2 y^{'} (x)}{x} \\ y^{''} (x) = \frac{- x y (x) - 2 y^{'} (x)}{x} \\ y^{''} (0) = 0 \end{matrix}$

Hence, it satisfies L'Hospital rule and by applying it in differential equation it becomes,

$\begin{matrix} y^{''} (x) = \frac{- x y (x) - 2 y^{'} (x)}{x} \\ y^{''} (x) = - 2 y^{''} (x) - y (x) - x y^{'} (x) \\ y^{''} (0) = - 2 y^{''} (0) - y (0) - x y^{'} (0) \\ y^{''} (0) = \frac{- 1}{3} \end{matrix}$

Step 2:Final proof

The Taylor polynomial of degree 2 in the solution is given by. $p_{3} (x) = 1 - \frac{x^{2}}{6}$

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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

Step by Step Solution

Step 1:To Find the Taylor polynomial of degree

Step 2:Final proof

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

The solution to the initial value problemxy''(x)+2y'(x)+xy(x)=0y(0)=1, y'(0)=0has derivatives of all orders at x=0 (although this is far from obvious). Use L'Hôpital's rule to compute the Taylor polynomial of degree 2 approximating this solution.

Step by Step Solution

Step 1:To Find the Taylor polynomial of degree

Step 2:Final proof

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