Calculus I, Fall 2011 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 13

Assigned Dec 08, 2011.

1. Section 6.11: Problems: 17, 21, 23, 27.

2. Verify that both sinh x are cosh x are solutions of y′′ = y and then solve for y′′ = y, y(0) = 1, y′(0) = 2.

3. Verify that both e^{2x} and xe^{2x} are solutions of y′′ − 4y′ + 4y = 0, therefore so is the combination a_{1}e^{2x} + a_{2}xe^{2x}. This is an example of the multiple root case: (λ − 2)^{2} = 0. You can either verify by direct differentiation, or try to look for solutions of the form z(x)e^{2x} and find that this leads to z′′ = (y(x)e^{−2x})′′ = 0.

4. Verify by direct differentiation that

\frac{d}{dx}(cos(kx) + i sin(kx)) = ik(cos(kx) + i sin(kx))

This is a good explanation why one defines exp(ikx) to be cos(kx) + i sin(kx).

5. Section 7.1: Problems: 9, 37, 47.

6. Section 7.2: Problems: 3, 5, 41.

7. Section 7.3: Problems: 43, 45, 46.