Chapter 7: Second-Order Differential Equations

boundary conditions

the conditions that give the state of a system at different times, such as the position of a spring-mass system at two different times

boundary-value problem

a differential equation with associated boundary conditions

characteristic equation

the equation 𝑎⁢𝜆²+𝑏⁢𝜆+𝑐=0 for the differential equation 𝑎⁢𝑦″+𝑏⁢𝑦′+𝑐⁢𝑦=0

complementary equation

homogeneous linear equation

a second-order differential equation that can be written in the form 𝑎2⁡(𝑥)⁢𝑦″+𝑎1⁡(𝑥)⁢𝑦′+𝑎0⁡(𝑥)⁢𝑦=𝑟⁡(𝑥), but 𝑟⁡(𝑥)=0 for every value of 𝑥

linearly dependent

a set of functions 𝑓1⁡(𝑥),𝑓2⁡(𝑥)⁢,…,⁢𝑓𝑛⁡(𝑥) for which there are constants 𝑐1,𝑐2⁢,…⁢𝑐𝑛, not all zero, such that 𝑐1⁢𝑓1⁡(𝑥)+𝑐2⁢𝑓2⁡(𝑥)+⋯+𝑐𝑛⁢𝑓𝑛⁡(𝑥)=0 for all x in the interval of interest

linearly independent

a set of functions 𝑓1⁡(𝑥),𝑓2⁡(𝑥)⁢,…,⁢𝑓𝑛⁡(𝑥) for which there are no constants 𝑐1,𝑐2⁢,…⁢𝑐𝑛, such that 𝑐1⁢𝑓1⁡(𝑥)+𝑐2⁢𝑓2⁡(𝑥)+⋯+𝑐𝑛⁢𝑓𝑛⁡(𝑥)=0 for all x in the interval of interest

method of undetermined coefficients

a method that involves making a guess about the form of the particular solution, then solving for the coefficients in the guess

method of variation of parameters

a method that involves looking for particular solutions in the form 𝑦𝑝⁡(𝑥)=𝑢⁡(𝑥)⁢𝑦1⁡(𝑥)+𝑣⁡(𝑥)⁢𝑦2⁡(𝑥), where 𝑦1 and 𝑦2 are linearly independent solutions to the complementary equations, and then solving a system of equations to find 𝑢⁡(𝑥) and 𝑣⁡(𝑥)

nonhomogeneous linear equation

a second-order differential equation that can be written in the form 𝑎2⁡(𝑥)⁢𝑦″+𝑎1⁡(𝑥)⁢𝑦′+𝑎0⁡(𝑥)⁢𝑦=𝑟⁡(𝑥), but 𝑟⁡(𝑥)≠0 for some value of 𝑥

particular solution

a solution 𝑦𝑝⁡(𝑥) of a differential equation that contains no arbitrary constants

RLC series circuit

a complete electrical path consisting of a resistor, an inductor, and a capacitor; a second-order, constant-coefficient differential equation can be used to model the charge on the capacitor in an RLC series circuit

simple harmonic motion

motion described by the equation 𝑥⁡(𝑡)=𝑐1⁢cos⁡(𝜔⁢𝑡)+𝑐2⁢sin⁡(𝜔⁢𝑡), as exhibited by an undamped spring-mass system in which the mass continues to oscillate indefinitely

steady-state solution

a solution to a nonhomogeneous differential equation related to the forcing function; in the long term, the solution approaches the steady-state solution

Linear second-order differential equation

Second-order equation with constant coefficients

General solution to a nonhomogeneous linear differential equation

Equation of simple harmonic motion

Solution for simple harmonic motion

Alternative form of solution for SHM

Forced harmonic motion

Charge in a RLC series circuit