Short Answer

Find a general solution for the differential equation with x as the independent variable:
$y^{(4)} + 4 y'' + 4 y = 0$

The general solution for the differential equation with x as the independent variable is

$y (x) = c_{1} \cos (\sqrt{2} x) + c_{2} x \cos (\sqrt{2} x) + c_{3} \sin (\sqrt{2} x) + c_{4} x \sin (\sqrt{2} x)$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Auxiliary equation:

The auxiliary equation in this problem is $r^{4} + 4 r^{2} + 4 = 0$ . This can be factored as ${(r^{2} + 2)}^{2}$ =0. Therefore this equation has roots $r = \sqrt{2} i, - \sqrt{2} i, \sqrt{2} i, - \sqrt{2} i$ , which we see are repeated and complex.

Step 2: General solution:

The general solution to the given equation is given by

$y (x) = c_{1} \cos (\sqrt{2} x) + c_{2} x \cos (\sqrt{2} x) + c_{3} \sin (\sqrt{2} x) + c_{4} x \sin (\sqrt{2} x)$

Hence the final solution is $y (x) = c_{1} \cos (\sqrt{2} x) + c_{2} x \cos (\sqrt{2} x) + c_{3} \sin (\sqrt{2} x) + c_{4} x \sin (\sqrt{2} x)$

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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

Find a general solution for the differential equation with x as the independent variable:
$y^{(4)} + 4 y'' + 4 y = 0$

Step by Step Solution

Step 1: Auxiliary equation:

Step 2: General solution:

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Find a general solution for the differential equation with x as the independent variable: y(4)+4y''+4y=0

Step by Step Solution

Step 1: Auxiliary equation:

Step 2: General solution:

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Find a general solution for the differential equation with x as the independent variable:
$y^{(4)} + 4 y'' + 4 y = 0$