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 mass–spring system is driven by a sinusoidal external force $g (t) = 5 sint$ . The mass equals 1, the spring constant equals 3, and the damping coefficient equals 4. If the mass is initially located at $y (0) = \frac{1}{2}$ and at rest, i.e., $y' (0) = 0$ , find its equation of motion.

The equation of motion is $y = 2 e^{- t} - \frac{1}{2} e^{- 3 t} - cost + \frac{1}{2} sint .$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Use the given information for writing the differential equation.

Given that,

The mass equals $m = 1$ ,

The spring constant equals $c = 3$ ,

And the damping coefficient equals $b = 4$ .

The differential equation is,

$\begin{matrix} my'' + by' + cy = g (t) \\ y'' + 4 y' + 3 y = 5 sint . ..... (1) \end{matrix}$

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

$y'' + 4 y' + 3 y = 0$

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

The auxiliary equation for the above equation,

$m^{2} + 4 m + 3 = 0$

Solve the auxiliary equation,

$\begin{matrix} m^{2} + 4 m + 3 = 0 \\ m^{2} + 3 m + m + 3 = 0 \\ m (m + 3) + 1 (m + 3) = 0 \\ (m + 1) (m + 3) = 0 \end{matrix}$

The roots of auxiliary equation are,

$m_{1} = - 1, m_{2} = - 3$

The complimentary solution of the given equation is,

$y_{c} = c_{1} e^{- t} + c_{2} e^{- 3 t}$

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

Assume, the particular solution of equation (1),

$y_{p} (t) = Acost + Bsint . ..... (2)$

Now find the first and second derivative of above equation,

$\begin{matrix} y_{p}' (t) = - Asint + Bcost \\ y_{p}'' (t) = - Acost - Bsint \end{matrix}$

Substitute the value of $y_{p} (t), y_{p}' (t)$ and $y_{p}'' (t)$ in the equation (1),

$\begin{matrix} y'' + 4 y' + 3 y = 5 sint \\ - Acost - Bsint + 4 (- Asint + Bcost) + 3 (Acost + Bsint) = 5 sint \\ (2 B - 4 A) sint + (2 A + 4 B) cost = 5 sint \end{matrix}$

Comparing the all coefficients of the above equation,

$\begin{matrix} 2 B - 4 A = 5 . ...... (3) \\ 4 B + 2 A = 0 . ..... (4) \end{matrix}$

Solve the above equations,

$\begin{matrix} 2 (2 B - 4 A) = 5 \times 2 \\ 4 B - 8 A = 10 \\ 4 B + 2 A = 0 \\ A = - 1 \end{matrix}$

Substitute the value of A in the equation (4),

role="math" localid="1655115721256" $\begin{matrix} 4 B + 2 (- 1) = 0 \\ B = \frac{1}{2} \end{matrix}$

Therefore, the particular solution of equation (1),

$\begin{matrix} y_{p} (t) = Acost + Bsint \\ y_{p} (t) = - cost + \frac{1}{2} sint \end{matrix}$

Step 4: Find the general solution and use the given initial condition,

Therefore, the general solution is,

$\begin{array}{l} y = y_{c} (t) + y_{p} (t) \\ y = c_{1} e^{- t} + c_{2} e^{- 3 t} - cost + \frac{1}{2} sint . ..... (5) \end{array}$

Given initial condition,

$y (0) = \frac{1}{2}, y' (0) = 0$

Substitute the value of $y = \frac{1}{2}$ and t = 0 in the equation (3),

$\begin{matrix} \frac{1}{2} = c_{1} e^{- (0)} + c_{2} e^{- 3 (0)} - \cos (0) + \frac{1}{2} \sin (0) \\ \frac{1}{2} = c_{1} + c_{2} - 1 \\ c_{1} + c_{2} = \frac{3}{2} . .... (6) \end{matrix}$

Now find the derivative of above equation,

$y' = - c_{1} e^{- t} - 3 c_{2} e^{- 3 t} + sint + \frac{1}{2} cost$

Substitute the value of y’ = 0 and t = 0 in the above equation,

$\begin{matrix} 0 = - c_{1} e^{- (0)} - 3 c_{2} e^{- 3 (0)} + \sin (0) + \frac{1}{2} \cos (0) \\ 0 = - c_{1} - 3 c_{2} + \frac{1}{2} \\ c_{1} + 3 c_{2} = \frac{1}{2} . ..... (7) \end{matrix}$

Solve the (6) and (7) equations,

$\begin{matrix} c_{1} + c_{2} = \frac{3}{2} \\ c_{1} + 3 c_{2} = \frac{1}{2} \\ c_{2} = - \frac{1}{2} \end{matrix}$

Substitute the value of $c_{2}$ in the equation (6),

role="math" localid="1655116289545" $\begin{matrix} c_{1} + c_{2} = \frac{3}{2} \\ c_{1} + (\frac{- 1}{2}) = \frac{3}{2} \\ c_{1} = 2 \end{matrix}$

Substitute the value of $c_{1}$ and $c_{2}$ in the equation (5),

role="math" localid="1655116374807" $\begin{matrix} y = c_{1} e^{- t} + c_{2} e^{- 3 t} - cost + \frac{1}{2} sint \\ y = 2 e^{- t} - \frac{1}{2} e^{- 3 t} - cost + \frac{1}{2} sint \end{matrix}$

Thus, the equation of motion is:

$y = 2 e^{- t} - \frac{1}{2} e^{- 3 t} - cost + \frac{1}{2} sint$

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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

A mass–spring system is driven by a sinusoidal external force $g (t) = 5 sint$ . The mass equals 1, the spring constant equals 3, and the damping coefficient equals 4. If the mass is initially located at $y (0) = \frac{1}{2}$ and at rest, i.e., $y' (0) = 0$ , find its equation of motion.

Step by Step Solution

Step 1: Use the given information for writing the differential equation.

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

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

Step 4: Find the general solution and use the given initial condition,

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

A mass–spring system is driven by a sinusoidal external force g(t)=5sint. The mass equals 1, the spring constant equals 3, and the damping coefficient equals 4. If the mass is initially located at y(0)=12and at rest, i.e., y'(0)=0, find its equation of motion.

Step by Step Solution

Step 1: Use the given information for writing the differential equation.

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

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

Step 4: Find the general solution and use the given initial condition,

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A mass–spring system is driven by a sinusoidal external force $g (t) = 5 sint$ . The mass equals 1, the spring constant equals 3, and the damping coefficient equals 4. If the mass is initially located at $y (0) = \frac{1}{2}$ and at rest, i.e., $y' (0) = 0$ , find its equation of motion.