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. $2 y'' (x) - 6 y' (x) + y (x) = \frac{sinx}{e^{4 x}}$

Yes, the method of undetermined coefficients can be applied to find a particular solution of the given equation.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Use the method of undetermined coefficients

Given equation,

$2 y'' (x) - 6 y' (x) + y (x) = \frac{\sin x}{e^{4 x}} . .... (1)$

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

$2 y'' (x) - 6 y' (x) + y (x) = 0$

The auxiliary equation for the above equation,

$2 m^{2} - 6 m + 1 = 0$

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

Solve the auxiliary equation,

$\begin{matrix} 2 m^{2} - 6 m + 1 = 0 \\ m = \frac{- (- 6) \pm \sqrt{36 - 4 (2) (1)}}{2 (2)} \\ m = \frac{6 \pm \sqrt{28}}{4} \\ m = \frac{3 \pm \sqrt{7}}{2} \end{matrix}$

The roots of the auxiliary equation are,

$m_{1} = \frac{3 + \sqrt{7}}{2}, m_{2} = \frac{3 - \sqrt{7}}{2}$

The complementary solution of the given equation is,

$y_{c} (x) = e^{\frac{3}{2} x} [c_{1} \cos (\frac{\sqrt{7}}{2}) + c_{2} \sin (\frac{\sqrt{7}}{2})]$

Step 3: Final conclusion.

According to the method of undetermined coefficients,

If $β \neq 0$ , then the blow equation has a particular solution,

$ay'' (x) + by' (x) + cy (x) = {Ct}^{m} e^{αx} sinβx$

Compare with the given differential equation,

$2 y'' (x) - 6 y' (x) + y (x) = e^{- 4 x} sinx$

We have,

$α = - 4, β = 1$

Condition satisfies,

s = 0 if $α + iβ$ is not a root of the associated auxiliary equation;

The roots of the auxiliary equation are different,

Therefore, s=0

The particular solution of the equation,

$y_{p} (x) = t^{s} (A_{m} t^{m} + . .. + A_{1} t + A_{0}) e^{αx} cosβx + t^{s} (B_{m} t^{m} + . .. + B_{1} t + B_{0}) e^{αx} sinβx$

Therefore, the method of undetermined coefficients can be applied.

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Fundamentals Of Differential Equations And Boundary Value Problems

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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. $2 y'' (x) - 6 y' (x) + y (x) = \frac{sinx}{e^{4 x}}$

Step by Step Solution

Step 1: Use the method of undetermined coefficients

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

Step 3: Final conclusion.

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Decide whether or not the method of undetermined coefficients can be applied to find a particular solution of the given equation.2y''(x)-6y'(x)+y(x)=sinxe4x

Step by Step Solution

Step 1: Use the method of undetermined coefficients

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

Step 3: Final conclusion.

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Decide whether or not the method of undetermined coefficients can be applied to find a particular solution of the given equation. $2 y'' (x) - 6 y' (x) + y (x) = \frac{sinx}{e^{4 x}}$