Higher order linear equations: existence and uniqueness of solutions;
independence of solutions, Wronskian; structure of the general solution to
a homogeneous/nonhomogeneous equation; operator notation
Homogeneous linear equations with constant coefficients
Nonhomogeneous equations: method of undetermined coefficients
Nonhomogeneous equations: variation of parameters
Reduction of order
Applications of second order linear O.D.E. (pay special attention
to harmonic oscillators, such as spring, oscillating circuit, simple pendulum). You can use formulas for damped and undamped vibration; however, in certain cases (say, forced vibration) considering the equation might still be necessary (see, e.g., Problem 1 in 1999 Final)
Nonlinear equations: reduction of order (see 16.8, 16.9)