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

A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation. $y'' = 2 y + 2 \tan^{3} x, y_{p} (x) = tanx$

The general solution of the given differential equation is $y = c_{1} e^{\sqrt{2} x} + c_{2} e^{- \sqrt{2} x} + tanx$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Firstly, write the auxiliary equation of the given differential equation

The differential equation is,

$\begin{matrix} y'' = 2 y + 2 \tan^{3} x \\ y'' - 2 y = 2 \tan^{3} x \dots (1) \end{matrix}$

Write the homogeneous differential equation of the equation (1),

$y'' - 2 y = 0$

The auxiliary equation for the above equation,

$m^{2} - 2 = 0$

Step 2: Now find the complementary solution of the given equation is

Solve the auxiliary equation,

$\begin{matrix} m^{2} - 2 = 0 \\ m = \pm \sqrt{2} \end{matrix}$

The roots of the auxiliary equation are,

$m_{1} = \sqrt{2}, & m_{2} = - \sqrt{2}$

The complementary solution of the given equation is,

$y_{c} = c_{1} e^{\sqrt{2} x} + c_{2} e^{- \sqrt{2} x}$

Step 3: Use the given particular solution to find a general solution for the equation.

The given particular solution,

$y_{p} (x) = tanx$

Therefore, the general solution is,

$\begin{matrix} y = y_{c} (x) + y_{p} (x) \\ y = c_{1} e^{\sqrt{2} x} + c_{2} e^{- \sqrt{2} x} + tanx \end{matrix}$

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation. $y'' = 2 y + 2 \tan^{3} x, y_{p} (x) = tanx$

Step by Step Solution

Step 1: Firstly, write the auxiliary equation of the given differential equation

Step 2: Now find the complementary solution of the given equation is

Step 3: Use the given particular solution to find a general solution for the equation.

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation. y''=2y+2tan3x, yp(x)=tanx

Step by Step Solution

Step 1: Firstly, write the auxiliary equation of the given differential equation

Step 2: Now find the complementary solution of the given equation is

Step 3: Use the given particular solution to find a general solution for the equation.

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A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation. $y'' = 2 y + 2 \tan^{3} x, y_{p} (x) = tanx$