Homework 1, due September 2, 2015

[goodwine]

Reading:

• Read sections 1.1 - 1.4.
• You are responsible for the material in section 1.5, but can read it at your convenience as long as it is before the end of September.
• Reading for section 1.6 will be assigned separately.

Exercises: do problem 1.8. In addition to identifying the attributes of the equation as required by the problem, also change one term in the equation in a manner that changes at least one of the attributes. Indicate which attribute(s) was changed. Also change one of the terms in a manner that does not change any of the attributes of the equation. Indicate the change and why it changed none of the attributes.

By "attributes" I mean

1. ordinary or partial
2. order
3. linear or nonlinear
4. if linear, homogeneous or inhomogeneous
5. if linear, constant or variable coefficient.

[goodwine]

Someone asked me:

For Homework one when I give a reason why I didn't change any attributes, what type of explanation are you looking for? Should I go attribute by attribute and give a reason for each one or is something like "changing sin(t) to cos(t) has no effect on any properties that would change the attributes" acceptable?

It's ok to focus on the "closest" attribute that could be changed by your modification. For example changing

x' + 3 x = 4

to

x' + 4 x = 4

Changes only a coefficient of one of the terms. You can answer "I changed the 3 to a 4, but the eqn is still constant coefficient because 4 is a constant like 3. All the other attributes obviously remain unchanged."

[goodwine]

Someone asked me:

For the differential equations in problem 1.8, before we classify the attributes of each equation, do we input the additional information that is included next to each equation first?
Like, for example, equation #2: 5x'' + 6x' + sin(t)x = cos(t) also has x(0) = 1, x'(0) = pi.
Do we include that information first, changing equation #2 to 5x'' + 6pi =1 or 5x'' = 6pi + 1 (is it alright to move them around?), and then find the attributes or do we ignore the extra information?

Because the initial conditions are not part of the definition of any of the attributes, you do not substitute them into the equation before determining the attributes.

[goodwine]

Someone asked me:

If "t" is not in the differential equation and the additional information is only "x(0)=1", is it correct to say that the independent variable is "t"? Is it more correct to say that the independent variable could be represented by any letter because the equation is not describing specifically a time-derivative?

If there is dot notation for the derivative, that generally connotes that the derivative is with respect to time.

[ldillon1]

Professor Goodwine,

On this homework you mention needing to know the material in 1.5 by the end of September, does that include Tuesday's exam?
I'm sorry if I didn't catch us discussing this in class.

Thank you!!

[goodwine]

Professor Goodwine,

On this homework you mention needing to know the material in 1.5 by the end of September, does that include Tuesday's exam?
I'm sorry if I didn't catch us discussing this in class.

Thank you!!

You certainly need to be able to apply Newton's law. But the level expected is like the home work problems, not the hovercraft or bead on the wire problems in the book.