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.

Hey, since there are multiple questions posted, we will answer the first question. If you want any…

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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