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 to the given equation. $5 y'' - 3 y' + 2 y = t^{3} \cos 4 t$

Yes, the method of undetermined coefficients can be applied.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Use the method of undetermined coefficients to find a particular solution of given differential equation.

Given equation,

$5 y'' - 3 y' + 2 y = t^{3} \cos 4 t \dots (1)$

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

$5 y'' - 3 y' + 2 y = 0$

The auxiliary equation for the above equation,

$5 m^{2} - 3 m + 2 = 0$

Step 2: Now find the roots of an auxiliary equation,

Solve the auxiliary equation,

$\begin{matrix} 5 m^{2} - 3 m + 2 = 0 \\ m = \frac{- (- 3) \pm \sqrt{9 - 4 (5) (2)}}{2 (5)} \\ m = \frac{3 \pm \sqrt{- 31}}{10} \\ m = \frac{3 \pm i \sqrt{31}}{10} \end{matrix}$

The roots of the auxiliary equation are,

$m_{1} = \frac{3 + i \sqrt{31}}{10}, m_{2} = \frac{3 - i \sqrt{31}}{10}$

The complementary solution of the given equation is,

$y_{c} (x) = e^{\frac{3}{10} x} [c_{1} \cosh (\frac{\sqrt{31}}{10}) + c_{2} \sinh (\frac{\sqrt{31}}{10})]$

Step 3: Final conclusion:

According to the method of undetermined coefficients,

The method of undetermined coefficients applies only to non-homogeneities that are polynomials, exponentials, sines, or cosines, or products of these functions, and R.H.S. of the differential equation has a finite family.

And the given R.H.S. of the equation $t^{3} \cos (4 t)$ has a final family.

The particular solution of equation (1),

$y_{p} (x) = ({At}^{3} + {Bt}^{2} + Ct + D) \cos (4 t) + ({Et}^{3} + {Ft}^{2} + Gt + H) \sin (4 t)$

Therefore, the method of undetermined coefficients can be applied.

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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 to the given equation. $5 y'' - 3 y' + 2 y = t^{3} \cos 4 t$

Step by Step Solution

Step 1: Use the method of undetermined coefficients to find a particular solution of given differential equation.

Step 2: Now find the roots of an 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 to the given equation.5y''-3y'+2y=t3cos4t

Step by Step Solution

Step 1: Use the method of undetermined coefficients to find a particular solution of given differential equation.

Step 2: Now find the roots of an 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 to the given equation. $5 y'' - 3 y' + 2 y = t^{3} \cos 4 t$