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Q6E

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Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 180

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

Decide whether or not the method of undetermined coefficients can be applied to find a particular solution of the given equation. $2 ω'' (x) - 3 ω (x) = 4 {xsin}^{2} x + 4 {xcos}^{2} x$

Yes, the method of undetermined coefficients can be applied.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Simplification of the given differential equation.

Given equation,

$2 ω'' (x) - 3 ω (x) = 4 {xsin}^{2} x + 4 {xcos}^{2} x$

Simplify the above equation,

$\begin{array}{l} 2 ω'' (x) - 3 ω (x) = 4 x (\sin^{2} x + \cos^{2} x) \\ 2 ω'' (x) - 3 ω (x) = 4 x . ..... (1) \end{array}$

Step 2: Now find the roots of the auxiliary equation.

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

$2 ω'' (x) - 3 ω (x) = 0$

The auxiliary equation for the above equation,

role="math" localid="1654859563585" $\begin{matrix} 2 m^{2} - 3 = 0 \\ 2 m^{2} = 3 \\ m = \pm \frac{3}{2} \end{matrix}$

The roots of the auxiliary equation are,

role="math" localid="1654859598798" $m_{1} = \frac{3}{2}, m_{2} = - \frac{3}{2}$

The complementary solution of the given equation is,

$ω_{c} (x) = c_{1} e^{\frac{3}{2} x} + c_{2} e^{- \frac{3}{2} x}$ .

Step 3: Final conclusion

The R.H.S. of equation is (4x).

Therefore, the particular solution of the equation,

$y_{p} (x) = Ax + b$

So, the method of undetermined coefficients can be applied.

Most popular questions for Math Textbooks

Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equation. Do not solve the equation. $2 y'' - y' + 6 y = t^{2} e^{- t} sint - 8 tcos 3 t + 10^{t}$

Find a particular solution to the given higher-order equation. $y''' + y'' - 2 y = {te}^{t} + 1$

A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation. $y'' - y = t, y_{p} (t) = - t$

Swinging Door. The motion of a swinging door with an adjustment screw that controls the amount of friction on the hinges is governed by the initial value problem

$Iθ'' + bθ' + kθ = 0; θ (0) = θ_{0}, θ' (0) = v_{0}$ ,

where $θ$ is the angle that the door is open, $I$ is the moment of inertia of the door about its hinges, $b > 0$ is a damping constant that varies with the amount of friction on the door, $k > 0$ is the spring constant associated with the swinging door, $θ_{0}$ is the initial angle that the door is opened, and $v_{0}$ is the initial angular velocity imparted to the door (see figure). If $I$ and $k$ are fixed, determine for which values of $b$ the door will not continually swing back and forth when closing.

Find a particular solution to the differential equation.

$y'' + 2 y' + 4 y = 111 e^{2 t} \cos 3 t$

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