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

Find a particular solution to the differential equation.
$y'' - y' + 9 y = 3 \sin 3 t$

The particular solution of the given differential equation is $y_{p} = \cos (3 t) .$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Firstly, write the auxiliary equation of the above differential equation.

The differential equation is:

$y'' - y' + 9 y = 3 \sin 3 t . ..... (1)$

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

$y'' - y' + 9 y = 0$

The auxiliary equation for the above equation,

$m^{2} - m + 9 = 0$

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

Solve the auxiliary equation,

$\begin{matrix} m^{2} - m + 9 = 0 \\ m = \frac{- (- 1) \pm \sqrt{1 - 4 (9) (1)}}{2} \\ m = \frac{1 \pm \sqrt{- 35}}{2} \\ m = \frac{1 \pm i \sqrt{35}}{2} \end{matrix}$

The roots of the auxiliary equation are,

$m_{1} = \frac{1 + i \sqrt{35}}{2}, & m_{2} = \frac{1 - i \sqrt{35}}{2}$

The complementary solution of the given equation is,

$y_{c} = e^{\frac{t}{2}} c_{1} \cos (\frac{\sqrt{35}}{2} t) + c_{2} \sin (\frac{\sqrt{35}}{2} t)$

Step 3: Use the method of undetermined coefficients to find a particular solution to the differential equation

According to the method of undetermined coefficients, assume the particular solution of equation (1),

$y_{p} = Asin (3 t) + Bcos (3 t) \dots (2)$

Now find the derivative of the above equation,

$\begin{matrix} y_{p}' = 3 Acos (3 t) - 3 Bsin (3 t) \\ y_{p}'' = - 9 Asin (3 t) - 9 Bcos (3 t) \end{matrix}$

From the equation (1), Substitute the value of $y_{p}'', y_{p}'$ and $y_{p}$ in the equation (1),

$\begin{matrix} y_{p}'' - y_{p}' + 9 y_{p} = 3 \sin 3 t \\ - 9 Asin (3 t) - 9 Bcos (3 t) - (3 Acos (3 t)) + 9 (+ Bcos (3 t)) = 3 \sin 3 t \\ (- 9 A + 3 B + 9 A) \sin (3 t) + (- 9 B - 3 A + 9 B) \cos (3 t) = 3 \sin 3 t \\ 3 Bsin (3 t) - 3 Acos (3 t) = 3 \sin 3 t \end{matrix}$

Step 4: Final conclusion.

Comparing all coefficients of the above equation;

$\begin{matrix} 3 B = 3 \\ B = 1 \\ - 3 A = 0 \\ A = 0 \end{matrix}$

Therefore, the particular solution of equation (1),

$\begin{matrix} y_{p} = Asin (3 t) + Bcos (3 t) \\ y_{p} = (0) \sin (3 t) + (1) \cos (3 t) \\ y_{p} = \cos (3 t) \end{matrix}$

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Find a particular solution to the differential equation.
$y'' - y' + 9 y = 3 \sin 3 t$

Step by Step Solution

Step 1: Firstly, write the auxiliary equation of the above differential equation.

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

Step 3: Use the method of undetermined coefficients to find a particular solution to the differential equation

Step 4: Final conclusion.

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Find a particular solution to the differential equation.y''-y'+9y=3sin3t

Step by Step Solution

Step 1: Firstly, write the auxiliary equation of the above differential equation.

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

Step 3: Use the method of undetermined coefficients to find a particular solution to the differential equation

Step 4: Final conclusion.

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Find a particular solution to the differential equation.
$y'' - y' + 9 y = 3 \sin 3 t$