Homework 2, due September 8, 2010.

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Reading: From the course text, sections 1.2-1.4 from Chapter 1 and all of Chapter 2.

Exercises: 2.2-2.8, 2.10 (in 2.10 only do numbers 1, 4, 10, 11 and 14) and 2.11.

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Someone asked me:

For problem 2.10, when you ask for the general solution using undetermined coefficients, do you want us to solve for the coefficients of sin(t), cos(t), and also e^lambda*t ? I don't see how we can solve for the coefficient of e^lambda*t without using a new variable x(0), which isn't given.

If the initial condition isn't given, then you just have an arbitrary constant multiplying the homogeneous solution. Its value would depend on x(0) (although it generally isn't equal to x(0)). One thing you said is incorrect, though, which is you don't solve for the coefficient of the exponential. The homogeneous and particular solutions are determined separately.

Also, will we be going over mixing problems in class on Monday? I am struggling with 2.6, and will wait to delve deeper into it if we are going over similar material on Monday.

No, I don't plan to. You'll just have to systematically set up the problem in a manner that let's you get the differential equation.

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Someone asked me:

I have some trouble with Problem 2.11. I feel like there is a special technique involved in the proof, but I can't figure it out. Could you please give me a hint on that problem?

You need to use the Wronskian. Due to the nature of the functions, the determinant has a special form that you can compute even though the dimension is an arbitrary n x n.