Bill Goodwine's Courses

Reading: Chapter 4.

Exercises: from the course text, problems 4.1, 4.3, 4.4, 4.5, 4.9 and 4.10.

Also, match the solutions to the following initial value problems to the plots. Every equation has a corresponding solution plot. For each equation, state the reason why the solution plot must be the corresponding plot. For example, an answer would be "the solution to equation 11 is the blue curve on plot 8 because..."

You may choose to solve every single differential equation in this homework. In that case I will be impressed, but only by your persistence.

Based only upon what we have covered in the class so far, it is possible to complete this homework by solving six differential equations. Doing so would require some creative thinking. I suspect that most people will have to solve eight or nine differential equations to complete this homework.

For solution plots 2, 3, 4, 6, 7 and 8 indicate what the difference is between the solutions graphed on the same plot and what the differences are in the corresponding equations. Construct another equation of a similar form and without actually solving it, sketch what its solution would approximately look like on the plot with the other three solutions.

Equations

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Solutions

1. red="a", blue="b", green="c"
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2. red="a", blue="b", green="c"
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3. red="a", blue="b", green="c"
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4. red="a", blue="b", green="c"
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5. red="a", blue="b", green="c"
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6. red="a", blue="b", green="c"
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7. red="a", blue="b", green="c"
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8. red="a", blue="b", green="c"
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9. red="a", blue="b", green="c"
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Hi Professor,
I just wanted to let you know that I believe there may be two typos in the book. On page 97, Figure 4.9 I believe that the caption should say "Forcing function and particular solution out of phase for w > wn" since the previous graph is the one that is in phase. Also on page 104, the particular solution is listed as the second derivative at the top of the page.
Thanks,
Maria Gaither

Dear Prof. Goodwine,
For eqn. 4.17, I believe the second term should contain the first derivative of x, not x

Professor,

In problem 4.9, is gravity in addition to the force on the mass, or is it the force on the mass?

Thanks

znussman wrote:Professor,

In problem 4.9, is gravity in addition to the force on the mass, or is it the force on the mass?

Gravity is in addition to the force on the mass, so add mg as a separate term.

For Problem 4.4, are the equations in the canonical form xdotdot +2zeta*wn xdot+wn^2 x = F/msinwt, as opposed to a xdotdot + b xdot +..., since the determination of the constants w_n, zeta, etc depends on what form is assumed

mkiener wrote:For Problem 4.4, are the equations in the canonical form xdotdot +2zeta*wn xdot+wn^2 x = F/msinwt, as opposed to a xdotdot + b xdot +..., since the determination of the constants w_n, zeta, etc depends on what form is assumed

If they aren't in canonical form, they should be put in it, but in this case there is nothing to do because the coefficient of the second derivative terms is 1 for every one of the equations.

Professor,
For problem 4.9, does a forcing function F(t) need to be given to solve the equation of motion?

smattix wrote:Professor,
For problem 4.9, does a forcing function F(t) need to be given to solve the equation of motion?

You don't need to solve it, only find the equation of motion for each of the two cases.

On Problem 4.9, if gravity is in addition to the force on the mass, what form should we assume for the force, since F(t) is not specified in either the diagram or the problem?

tmo3290 wrote:On Problem 4.9, if gravity is in addition to the force on the mass, what form should we assume for the force, since F(t) is not specified in either the diagram or the problem?

Sorry, didn't see that this question had already been answered.

Professor,

There is a small typo in equation 4.8, pg. 93 I believe; the "xdot" should be "xdot0".

Ben Rollin