1. In class, we talked about how to solve the homogeneous second-order differential equation ay" + by' + cy = 0 by looking at the roots of the corresponding characteristic equation. In particular, if we get two repeated roots, i.e., ri e"2", which are the same. Then we use this known solution to produce a second unknown solution by xe"1". This idea originates from a method called "Reduction of Order", is based on your previous experience on Calculus 1 and First-order differential equations you just learned in class. This problem will lead you to discover this idea in a little more depth. r2, we automatically get two particular solutions e"1", which (a) Consider the second-order differential equation y" + 4y' + 4y = 0. (1) What is the corresponding characteristic equation and the roots?

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