First Order Equations. Know how to solve first order linear equations using integrating factors (§1.2). Also, be able to solve separable equations (§1.4), and exact equations (§1.9). Expect an applications problem from §§1.3, 1.5, 1.7-1.8. From §1.10, know the the existence and uniqueness theorem, but you do not need to know how to estimate the interval of convergence. Concening the numerical methods in §1.13 (Euler) and §1.16 (Runge-Kutta), know what they are and be able to write a brief comparison of them.
Second Order Equations. We really are treating only linear second order equations of the form L[y]=g, where L is a linear operator. §2.1 covers the homogeneous case L[y]=0. Know how to find the Wronskian W, and be able to use it to tell whether a set of solutions is linearly independent (fundamental). Be able to derive W'+pW=0. §2.2 covers solving constant coefficient homogeneous equations, complex exponentials, and reduction of order. Be able to do problems similar to the ones in homework, including being able to find the polar form of a complex number. §2.3 is an observation: Two solutions of the nonhomogeneous equation L[y]=g differ by a solution to the homogeneous equation. Thus, if one knows a fundamental set {y1,y2} for L[y]=0 and a particular solution yp to L[y]=g, then y=yp + c1y1 + c2y2 is the the general solution to L[y]=g. Be able to use the method of variation of parameters to find yp; this is discussed in §2.4.
Comments. As I mentioned in class, when you use reduction of order or variation of parameters, always start from first principles. The combination of reduction of order and variation of parameters allows one to find the general solution to L[y]=g if we know just one solution to the homogeneous equation L[y]=0!