For these problems follow these instructions. #11,12,13,14 In each of Problems 11 through 14, find and plot both the steady periodic solution \( x_{sp}(t) = C \cos(\omega t - \alpha) \) of the given differential equation and the actual solution \( x(t) = x_{sp}(t) + x_{tr}(t) \) that satisfies the given initial conditions.1. **Problem 11**: \[ x'' + 4x' + 5x = 10 \cos 3t; \quad x(0) = x'(0) = 0 \]2. **Problem 12**: \[ x'' + 6x' + 13x = 10 \sin 5t; \quad x(0) = x'(0) = 0 \]3. **Problem 13**: \[ x'' + 2x' + 26x = 600 \cos 10t; \quad x(0) = 10, \quad x'(0) = 0 \]4. **Problem 14**: \[ x'' + 8x' + 25x = 200 \cos t + 520 \sin t; \quad x(0) = -30, \quad x'(0) = -10 \]Each of Problems 15 through 18 gives the parameters for a forced mass–spring–dashpot system with equation \( m x'' + c x' + k x = F_0 \cos \omega t \). Investigate the possibility of practical resonance in this system. In particular, find the amplitude \( C(\omega) \) of steady periodic forced oscillations as a function of frequency \( \omega \). Sketch the graph of \( C(\omega) \) and find the practical resonance frequency \( \omega \).5. **Problem 15**: \[ m = 1, \quad c = 2, \quad k = 2, \quad F_0 = 2 \]6. **Problem 16**: \[ m = 1, \quad c = 4, \quad k = 5, \quad F_0 = 10 \] For the problems in §5.6 you should follow these instructions:(i) Find \( x_{tr} \) and \( x_{sp} \).(ii) Put \( x_{sp} \) into amplitude/circular frequency/phase angle form.(iii) Find \( x_{sp} \) either by undetermined coefficients or by the EE method.(iv) You do NOT have to put \( x_{tr} \) into any special form.(v) You do NOT have to do any graphing.

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In each of Problems 11 through 14, find and plot both the steady periodic soluti x sp (t) = C cos(wt – a) of the given differential equation and the actual solut a(t) = xsp(t) +atr(t) that satisfies the given initial conditions. %3D 11. cll+4xl+ 5x = 10 cos 3t; x(0) = x/(0) = 0 %3D

In each of Problems 11 through 14, find and plot both the steady periodic soluti x sp (t) = C cos(wt – a) of the given differential equation and the actual solut a(t) = xsp(t) +atr(t) that satisfies the given initial conditions. %3D 11. cll+4xl+ 5x = 10 cos 3t; x(0) = x/(0) = 0 %3D

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

For these problems follow these instructions. #11,12,13,14

In each of Problems 11 through 14, find and plot both the steady periodic solution \( x_{sp}(t) = C \cos(\omega t - \alpha) \) of the given differential equation and the actual solution \( x(t) = x_{sp}(t) + x_{tr}(t) \) that satisfies the given initial conditions.

1. **Problem 11**:
\[
x'' + 4x' + 5x = 10 \cos 3t; \quad x(0) = x'(0) = 0
\]

2. **Problem 12**:
\[
x'' + 6x' + 13x = 10 \sin 5t; \quad x(0) = x'(0) = 0
\]

3. **Problem 13**:
\[
x'' + 2x' + 26x = 600 \cos 10t; \quad x(0) = 10, \quad x'(0) = 0
\]

4. **Problem 14**:
\[
x'' + 8x' + 25x = 200 \cos t + 520 \sin t; \quad x(0) = -30, \quad x'(0) = -10
\]

Each of Problems 15 through 18 gives the parameters for a forced mass–spring–dashpot system with equation \( m x'' + c x' + k x = F_0 \cos \omega t \). Investigate the possibility of practical resonance in this system. In particular, find the amplitude \( C(\omega) \) of steady periodic forced oscillations as a function of frequency \( \omega \). Sketch the graph of \( C(\omega) \) and find the practical resonance frequency \( \omega \).

5. **Problem 15**:
\[
m = 1, \quad c = 2, \quad k = 2, \quad F_0 = 2
\]

6. **Problem 16**:
\[
m = 1, \quad c = 4, \quad k = 5, \quad F_0 = 10
\]

For the problems in §5.6 you should follow these instructions:

(i) Find \( x_{tr} \) and \( x_{sp} \).

(ii) Put \( x_{sp} \) into amplitude/circular frequency/phase angle form.

(iii) Find \( x_{sp} \) either by undetermined coefficients or by the EE method.

(iv) You do NOT have to put \( x_{tr} \) into any special form.

(v) You do NOT have to do any graphing.

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We begin with the corresponding auxiliary equation.

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