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Q5E

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Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 341

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 1-6, use the method of variation of parameters to determine a particular solution to the given equation.
$y^{'''} + y^{'} = t a n x$

The particular solution is $y_{p} = \ln | \sec x | - \sin x \ln | \sec x + \tan x |$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition

Variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations.

Step 2: Find complementary solution

The given equation is: $y^{'''} + y^{'} = \tan x$

The auxiliary equation is $D^{3} + D = 0$

So, $D = \pm i, 0$

So ${1, \cos x, \sin x}$ fundamental set.

Step 3: Calculate Wornkians

The value of wronkians is:

$W [\begin{matrix} 1 & \cos x & s i n x \end{matrix}] = |\begin{matrix} 1 & \cos x & \sin x \\ 0 & - \sin x & \cos x \\ 0 & - \cos x & - \sin x \end{matrix}| = 1$

$W_{1} = (- 1)^{3 - 1} W [\begin{matrix} \cos x & \sin x \end{matrix}] = |\begin{matrix} \cos x & \sin x \\ - \sin x & \cos x \end{matrix}| = 1 W_{2} = (- 1)^{3 - 2} W [\begin{matrix} 1 & \sin x \end{matrix}] = - \cos x W_{3} = (- 1)^{3 - 3} W [\begin{matrix} 1 & \cos x \end{matrix}] = - \sin x$

Step 4: For particular solution

The particular solution is given by:

$y_{p} = 1 \int \frac{1 (\tan x)}{1} d x + \cos x \int \frac{- \cos x (\tan x)}{1} d x + \sin x \int \frac{- \sin x \cdot \tan x}{1} d x y_{p} = \ln | \sec x | + \cos^{2} x + \sin x I_{1} I_{1} = \int - \sin x \cdot \tan x d x = \sin x - \ln (\sec x + \tan x)$

$y_{p} = \ln | \sec x | + \cos^{2} x + \sin^{2} x - \sin^{2} x \ln (\sec x + \tan x)$

Since 1 is in fundamental set solution so $y_{p} = \ln | \sec x | - \sin x \ln | \sec x + \tan x |$

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