Covered today: We introduced the notions of vectors spaces and linear operators. We defined a special linear operator named D that helped us prove some theorems about the general solution to 2nd-order constant coefficient homogeneous DEs.

For next time: 1abcd, 7,8,9, and 12 from page 644 of the handout. Also, prove the following:

Suppose

$$r_1 = \alpha + \beta i \qquad \hbox{and} \qquad r_2 = \alpha - \beta i$$

and if C₁ and C₂ are arbitrary complex numbers. Then the function

y(t) = C₁e^{r₁ t} + C₂e^{r₂ t}

is real-valued if and only if C₁ and C₂ are conjugates.

(Note that you have to prove both directions here. That is, you have to show that if C₁ and C₂ are conjugates then the function is real-valued, and that if the function is real-valued, then C₁ and C₂ are conjugates.)