### Key Terms

- 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 $a{\lambda}^{2}+b\lambda +c=0$ for the differential equation $a{y}^{\u2033}+b{y}^{\prime}+cy=0$

- complementary equation
- for the nonhomogeneous linear differential equation

$${a}_{2}\left(x\right){y}^{\u2033}+{a}_{1}\left(x\right){y}^{\prime}+{a}_{0}\left(x\right)y=r\left(x\right),$$

the associated homogeneous equation, called the*complementary equation*, is

$${a}_{2}\left(x\right){y}^{\u2033}+{a}_{1}\left(x\right){y}^{\prime}+{a}_{0}\left(x\right)y=0$$

- homogeneous linear equation
- a second-order differential equation that can be written in the form ${a}_{2}\left(x\right){y}^{\u2033}+{a}_{1}\left(x\right){y}^{\prime}+{a}_{0}\left(x\right)y=r\left(x\right),$ but $r(x)=0$ for every value of $x$

- linearly dependent
- a set of functions ${f}_{1}(x),{f}_{2}(x)\text{,\u2026,}{f}_{n}(x)$ for which there are constants ${c}_{1},{c}_{2}\text{,\u2026}{c}_{n},$ not all zero, such that ${c}_{1}{f}_{1}(x)+{c}_{2}{f}_{2}(x)+\text{\cdots}+{c}_{n}{f}_{n}(x)=0$ for all
*x*in the interval of interest

- linearly independent
- a set of functions ${f}_{1}(x),{f}_{2}(x)\text{,\u2026,}{f}_{n}(x)$ for which there are no constants ${c}_{1},{c}_{2}\text{,\u2026}{c}_{n},$ such that ${c}_{1}{f}_{1}(x)+{c}_{2}{f}_{2}(x)+\text{\cdots}+{c}_{n}{f}_{n}(x)=0$ 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 ${y}_{p}(x)=u(x){y}_{1}(x)+v(x){y}_{2}(x),$ where ${y}_{1}$ and ${y}_{2}$ 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 ${a}_{2}\left(x\right){y}^{\u2033}+{a}_{1}\left(x\right){y}^{\prime}+{a}_{0}\left(x\right)y=r\left(x\right),$ but $r(x)\ne 0$ for some value of $x$

- particular solution
- a solution ${y}_{p}(x)$ 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 $x(t)={c}_{1}\text{cos}\phantom{\rule{0.1em}{0ex}}\left(\omega t\right)+{c}_{2}\text{sin}\phantom{\rule{0.1em}{0ex}}\left(\omega t\right),$ 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