System Dynamics
System Dynamics
3rd Edition
ISBN: 9780073398068
Author: III William J. Palm
Publisher: MCG
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Question
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Chapter 2, Problem 2.1P
To determine

(a)

Whether the given differential equation is linear or non-linear with supportive reason.

Expert Solution
Check Mark

Answer to Problem 2.1P

yy¨+5y˙+y=0 is a non-linear differential equation.

As, here, derivative of dependent variable is multiplied with itself.

Explanation of Solution

Given:

yy¨+5y˙+y=0.

Concept Used:

An ordinary differential for y=y(t) is said to be linear if it can be written in the form

an(t)yn+an1(t)yn1+...+a2(t)y2+a1(t)y+a0(t)y=g(t)

where the ‘coefficient’ functions an(t),an1(t),...,a2(t),a1(t),a0(t),g(t) can be any functions of t, (including the zero function).

In other words a differential equation is said to be linear if and only if the derivative of dependent variable should not be multiplied with dependent variable itself, otherwise it should be non-linear.

Calculation:

Given differential equation is

yy¨+5y˙+y=0

Which is a non-linear differential equation.

As, here, derivative of dependent variable is multiplied with itself.

To determine

(b)

Whether the given differential equation is linear or non-linear with supportive reason.

Expert Solution
Check Mark

Answer to Problem 2.1P

y˙+siny=0 is a linear differential equation.

As, here, derivative of dependent variable is not multiplied with itself.

Explanation of Solution

Given:

y˙+siny=0.

Concept Used:

An ordinary differential for y=y(t) is said to be linear if it can be written in the form

an(t)yn+an1(t)yn1+...+a2(t)y2+a1(t)y+a0(t)y=g(t)

where the ‘coefficient’ functions an(t),an1(t),...,a2(t),a1(t),a0(t),g(t) can be any functions of t, (including the zero function).

In other words a differential equation is said to be linear if and only if the derivative of dependent variable should not be multiplied with dependent variable itself, otherwise it should be non-linear.

Calculation:

Given differential equation is

y˙+siny=0 is a linear differential equation. As, here, derivative of dependent variable is not multiplied with itself.

To determine

(c)

Whether the given differential equation is linear or non-linear with supportive reason.

Expert Solution
Check Mark

Answer to Problem 2.1P

y˙+y=0 is a linear differential equation.

As, here, derivative of dependent variable is not multiplied with itself.

Explanation of Solution

Given:

y˙+y=0.

Concept Used:

An ordinary differential for y=y(t) is said to be linear if it can be written in the form

an(t)yn+an1(t)yn1+...+a2(t)y2+a1(t)y+a0(t)y=g(t)

where the ‘coefficient’ functions an(t),an1(t),...,a2(t),a1(t),a0(t),g(t) can be any functions of t, (including the zero function).

In other words a differential equation is said to be linear if and only if the derivative of dependent variable should not be multiplied with dependent variable itself, otherwise it should be non-linear.

Calculation:

Given differential equation is

y˙+y=0 is a linear differential equation.

As, here, derivative of dependent variable is not multiplied with itself.

To determine

(d)

Whether the given differential equation is linear or non-linear with supportive reason.

Expert Solution
Check Mark

Answer to Problem 2.1P

yy¨+5t2y˙+3y=0

is a non-linear differential equation.

As, here, derivative of dependent variable is multiplied with itself.

Explanation of Solution

Given:

yy¨+5t2y˙+3y=0.

Concept Used:

An ordinary differential for y=y(t) is said to be linear if it can be written in the form

an(t)yn+an1(t)yn1+...+a2(t)y2+a1(t)y+a0(t)y=g(t)

where the ‘coefficient’ functions an(t),an1(t),...,a2(t),a1(t),a0(t),g(t) can be any functions of t, (including the zero function).

In other words a differential equation is said to be linear if and only if the derivative of dependent variable should not be multiplied with dependent variable itself, otherwise it should be non-linear.

Calculation:

Given differential equation is

yy¨+5t2y˙+3y=0

Which is a non-linear differential equation.

As, here, derivative of dependent variable is multiplied with itself.

To determine

(e)

Whether the given differential equation is linear or non-linear with supportive reason.

Expert Solution
Check Mark

Answer to Problem 2.1P

y¨+3t2siny=0 is a non-linear differential equation.

As, here, derivative of dependent variable is multiplied with independent variable.

Explanation of Solution

Given:

y¨+3t2siny=0.

Concept Used:

An ordinary differential for y=y(t) is said to be linear if it can be written in the form

an(t)yn+an1(t)yn1+...+a2(t)y2+a1(t)y+a0(t)y=g(t)

where the ‘coefficient’ functions an(t),an1(t),...,a2(t),a1(t),a0(t),g(t) can be any functions of t, (including the zero function).

In other words a differential equation is said to be linear if and only if the derivative of dependent variable should not be multiplied with dependent variable itself, otherwise it should be non-linear.

Calculation:

Given differential equation is

y¨+3t2siny=0

Which is a non-linear differential equation.

As, here, function of dependent variable is multiplied with independent variable.

To determine

(f)

Whether the given differential equation is linear or non-linear with supportive reason.

Expert Solution
Check Mark

Answer to Problem 2.1P

y˙+ety=0 which is a linear differential equation.

As, here, function of dependent variable is not multiplied with independent variable.

Explanation of Solution

Given:

y˙+ety=0.

Concept Used:

An ordinary differential for y=y(t) is said to be linear if it can be written in the form

an(t)yn+an1(t)yn1+...+a2(t)y2+a1(t)y+a0(t)y=g(t)

where the ‘coefficient’ functions an(t),an1(t),...,a2(t),a1(t),a0(t),g(t) can be any functions of t, (including the zero function).

In other words a differential equation is said to be linear if and only if the derivative of dependent variable should not be multiplied with dependent variable itself, otherwise it should be non-linear.

Calculation:

Given differential equation is

y˙+ety=0

which is a linear differential equation.

As, here, function of dependent variable is not multiplied with independent variable.

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Chapter 2 Solutions

System Dynamics

Ch. 2 - Prob. 2.11PCh. 2 - Obtain the inverse Laplace transform xt for each...Ch. 2 - Solve the following problems: 5x=7tx0=3...Ch. 2 - Solve the following: 5x+7x=0x0=4 5x+7x=15x0=0...Ch. 2 - Solve the following problems: x+10x+21x=0x0=4x0=3...Ch. 2 - Solve the following problems: x+7x+10x=20x0=5x0=3...Ch. 2 - Solve the following problems: 3x+30x+63x=5x0=x0=0...Ch. 2 - Solve the following problems where x0=x0=0 ....Ch. 2 - Invert the following transforms: 6ss+5 4s+3s+8...Ch. 2 - Invert the following transforms: 3s+2s2s+10...Ch. 2 - Prob. 2.21PCh. 2 - Compare the LCD method with equation (2.4.4) for...Ch. 2 - Prob. 2.23PCh. 2 - Prob. 2.24PCh. 2 - (a) Prove that the second-order system whose...Ch. 2 - For each of the following models, compute the time...Ch. 2 - Prob. 2.27PCh. 2 - Prob. 2.28PCh. 2 - Prob. 2.29PCh. 2 - If applicable, compute , , n , and d for the...Ch. 2 - Prob. 2.31PCh. 2 - For each of the following equations, determine the...Ch. 2 - Prob. 2.33PCh. 2 - Obtain the transfer functions Xs/Fs and Ys/Fs for...Ch. 2 - a. Obtain the transfer functions Xs/Fs and Ys/Fs...Ch. 2 - Prob. 2.36PCh. 2 - Solve the following problems for xt . Compare the...Ch. 2 - Prob. 2.38PCh. 2 - Prob. 2.39PCh. 2 - Prob. 2.40PCh. 2 - Determine the general form of the solution of the...Ch. 2 - a. Use the Laplace transform to obtain the form of...Ch. 2 - Prob. 2.43PCh. 2 - Prob. 2.44PCh. 2 - Obtain the inverse transform in the form xt=Asint+...Ch. 2 - Use the Laplace transform to solve the following...Ch. 2 - Express the oscillatory part of the solution of...Ch. 2 - Prob. 2.48PCh. 2 - Prob. 2.49PCh. 2 - Prob. 2.50PCh. 2 - Prob. 2.51PCh. 2 - Prob. 2.52PCh. 2 - Prob. 2.53PCh. 2 - 2.54 The Taylor series expansion for tan t...Ch. 2 - 2.55 Derive the initial value theorem: Ch. 2 - Prob. 2.56PCh. 2 - Prob. 2.57PCh. 2 - Use MATLAB to obtain the inverse transform of the...Ch. 2 - Use MATLAB to obtain the inverse transform of the...Ch. 2 - Use MATLAB to solve for and plot the unit-step...Ch. 2 - Use MATLAB to solve for and plot the unit-impulse...Ch. 2 - Use MATLAB to solve for and plot the impulse...Ch. 2 - Use MATLAB to solve for and plot the response of...
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