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

In Problems 1–7, convert the given initial value problem into an initial value problem for a system in normal form.
$x''' - y = t; x (0) = x' (0) = x'' (0) = 4 2 x'' + 5 y'' - 2 y = 1; y (0) = y' (0) = 1$

$x'_{1} (t) = x_{2} (t) x'_{2} (t) = x_{3} (t) x'_{3} (t) = x_{4} (t) + t x'_{4} (t) = x_{5} (t) x'_{5} (t) = \frac{2 x_{4} (t) - 2 x_{3} (t) + 1}{5} x_{1} (0) = x_{2} (0) = x_{3} (0) = 4, x_{4} (0) = x_{5} (0) = 1$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Express equation in form of x

Here given $x''' - y = t$ and $2 x'' + 5 y'' - 2 y = 1 .$

Rewrite the equations $x''' = y + t$ and $x'' = \frac{- 5 y'' + 2 y + 1}{2}$

Denote,

$x_{1} (t) = x (t) x_{2} (t) = x' (t) x_{3} (t) = x'' (t) x_{4} (t) = y (t) x_{5} (t) = y' (t) x'_{1} (t) = x_{2} (t)$

The equation transforms as:

$x'_{1} (t) = x_{2} (t) x'_{2} (t) = x_{3} (t) x'_{3} (t) = x_{4} (t) + t x'_{4} (t) = x_{5} (t) x'_{5} (t) = \frac{2 x_{4} (t) - 2 x_{3} (t) + 1}{5}$

Step 2: The initial conditions

The given initial conditions are $x (0) = x' (0) = x'' (0) = 4$ and $y (0) = y' (0) = 1 .$

Initial conditions after transformations $x_{1} (0) = x_{2} (0) = x_{3} (0) = 4, x_{4} (0) = x_{5} (0) = 1 .$

This is the required result.

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

In Problems 1–7, convert the given initial value problem into an initial value problem for a system in normal form.
$x''' - y = t; x (0) = x' (0) = x'' (0) = 4 2 x'' + 5 y'' - 2 y = 1; y (0) = y' (0) = 1$

Step by Step Solution

Step 1: Express equation in form of x

Step 2: The initial conditions

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

In Problems 1–7, convert the given initial value problem into an initial value problem for a system in normal form.x'''-y=t;x(0)=x'(0)=x''(0)=42x''+5y''-2y=1;y(0)=y'(0)=1

Step by Step Solution

Step 1: Express equation in form of x

Step 2: The initial conditions

Most popular questions for Math Textbooks

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Pure Maths

In Problems 1–7, convert the given initial value problem into an initial value problem for a system in normal form.
$x''' - y = t; x (0) = x' (0) = x'' (0) = 4 2 x'' + 5 y'' - 2 y = 1; y (0) = y' (0) = 1$