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 $a{y}^{″}+b{y}^{\prime }+cy=0$

complementary equation

for the nonhomogeneous linear differential equation

$$a_2\left(x\right)y^{″}+a_1\left(x\right)y^{\prime }+a_0\left(x\right)y=r\left(x\right),$$

the associated homogeneous equation, called the complementary equation, is

$$a_2\left(x\right)y^{″}+a_1\left(x\right)y^{\prime }+a_0\left(x\right)y=0$$

homogeneous linear equation

a second-order differential equation that can be written in the form $a_2\left(x\right)y^{″}+a_1\left(x\right)y^{\prime }+a_0\left(x\right)y=r\left(x\right),$ but $r(x)=0$ for every value of $x$

linearly dependent

a set of functions $f_1(x),f_2(x)\text{,…,}{f}_n(x)$ for which there are constants $c_1,c_2\text{,…}{c}_n,$ not all zero, such that $c_1{f}_1(x)+c_2{f}_2(x)+\text{⋯}+c_n{f}_n(x)=0$ for all x in the interval of interest

linearly independent

a set of functions $f_1(x),f_2(x)\text{,…,}{f}_n(x)$ for which there are no constants $c_1,c_2\text{,…}{c}_n,$ such that $c_1{f}_1(x)+c_2{f}_2(x)+\text{⋯}+c_n{f}_n(x)=0$ 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 $a_2\left(x\right)y^{″}+a_1\left(x\right)y^{\prime }+a_0\left(x\right)y=r\left(x\right),$ 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 $x(t)=c_1\text{cos}\left(\omega t\right)+c_2\text{sin}\left(\omega t\right),$ 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