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Q4E

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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

In Problems 1–7, convert the given initial value problem into an initial value problem for a system in normal form.
$y^{6} (t) = {[y' (t)]}^{3} - \sin (y (t)) + e^{2 t}; y (0) = y' (0) = . . . . . . = y^{(5)} (0) = 0$

$x'_{6} (t) = {[x_{2} (t)]}^{3} - \sin (x_{1} (t)) + e^{2 t}$

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 $y^{6} (t) = {[y' (t)]}^{3} - \sin (y (t)) + e^{2 t}$

Denote,

$x_{1} (t) = y (t) x_{2} (t) = y' (t) x_{3} (t) = y'' (t) x_{4} (t) = y^{3} (t) x_{5} (t) = y^{4} (t) x_{6} (t) = y^{5} (t)$

The equation transforms as;

$x'_{1} (t) = x_{2} (t) x'_{2} (t) = y'' (t) = x_{3} (t) x'_{3} (t) = y''' (t) = x_{4} (t) x'_{4} (t) = x_{5} (t) x'_{5} (t) = x_{6} (t) x'_{6} (t) = {[x_{2} (t)]}^{3} - \sin (x_{1} (t)) + e^{2 t}$

Step 2: The initial conditions

The given initial conditions are $y (0) = y' (0) = . . . . . . = y^{(5)} (0) = 0 .$

Initial conditions after transformations $x_{1} (0) = x_{2} (0) = x_{3} (0) = x_{4} (0) = x_{5} (0) = x_{6} (0) = 0 .$

This is the required result.

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