Summary of Chapter 8 : Second Order Linear Differential Equations—Qualitative and Quantitative Aspects

In this chapter we look at the second order linear differential equation a2(t)x''+a1(t)x'+a0(t)x=0, where a2(t), a1(t), and a0(t) are continuous functions of t, and a2(t) is not equal to zero for a<t<b. We will develop several techniques for finding the general solution for special cases of a2(t)x''+a1(t)x'+a0(t)x=0.

We give an Oscillation Theorem that allows us to determine qualitative properties of solutions, such as oscillations and boundedness, even if we cannot find these solutions explicitly. The concepts of linear independence and linear dependence are introduced, which allow us to identify certain explicit solutions as general solutions. We develop additional methods for finding particular solutions of nonhomogeneous linear differential equations and discover that these methods, as well as those methods from Chapter 7, also apply to higher order linear differential equations. Explicit and numerical solutions of boundary value problems are also discussed in this chapter.

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