Ohio State University MATH 2255 practice-midterm-2.

You will be allowed to bring a single double-sided 8:5 × 11 page of notes for this midterm.
1. Decide if the following statements are TRUE or FALSE. You do NOT need to justify your answers.
(a) (1 point) Any periodic function with period T can be written in the form
f(t) = R cos 2Tπt + δ
for some constants R and δ.
2. Give examples of the following. Be as explicit as possible. You do NOT need to justify your answers.
(a) (2 points) Give an example of a piecewise continuous function on the interval [0; 5] which is not
continuous on the interval [0; 5].
3. (5 points) Find the smallest vector space of functions closed under differentiation which contains the function
f(t) = t2 sin 2t:
You do NOT need to justify your answer.
4. (5 points) Find the smallest vector space of functions closed under the linear operator t2D2 which contains
the function
f(t) = t4
You do NOT need to justify your answer.
5. (5 points) Find the values R and δ for which
cos t - 2 sin t = R cos(t - δ)
for all t 2 R.
6. Find general solutions for the following differential equations
(a) (5 points) y00 - 4y0 + 5y = t2