Summary of Second-Order Linear Equations

Second-order differential equations can be classified as linear or nonlinear, homogeneous or nonhomogeneous.
To find a general solution for a homogeneous second-order differential equation, we must find two linearly independent solutions. If [latex]y_{1}(x)[/latex] and [latex]y_{2}(x)[/latex] are linearly independent solutions to a second-order, linear, homogeneous differential equation, then the general solution is given by [latex]y(x)=c_{1}y_{1}(x)+c_{2}y_{2}(x)[/latex].
To solve homogeneous second-order differential equations with constant coefficients, find the roots of the characteristic equation. The form of the general solution varies depending on whether the characteristic equation has distinct, real roots; a single, repeated real root; or complex conjugate roots.
Initial conditions or boundary conditions can then be used to find the specific solution to a differential equation that satisfies those conditions, except when there is no solution or infinitely many solutions.

Key Equations

Linear second-order differential equation
[latex]a_{2}(x)y^{\prime\prime}+a_{1}(x)y^{\prime}+a_{0}(x)y=r(x)[/latex]
Second-order equation with constant coefficients
[latex]ay^{\prime\prime}+by^{\prime}+cy=0[/latex]

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 [latex]a\lambda^{2}+b\lambda+c=0[/latex] for the differential equation [latex]ay^{\prime\prime}+by^{\prime}+cy=0[/latex]

homogeneous linear equation: a second-order differential equation that can be written in the form [latex]a_{2}(x)y^{\prime\prime}+a_{1}(x)y^{\prime}+a_{0}(x)y=r(x)[/latex] but [latex]r(x)=0[/latex] for every value of [latex]x[/latex]

linearly dependent: a set of function [latex]f_{1}(x),f_{2}(x),\ldots f_{n}(x)[/latex] for which there are constants [latex]c_{1},c_{2},\ldots c_{n}[/latex], not all zero, such that [latex]c_{1}f_{1}(x) + c_{2}f_{2}(x) + {\cdots}+c_{n}f_{n}(x) = 0[/latex] for all [latex]x[/latex] in the interval of interest

linearly independent: a set of function [latex]f_{1}(x),f_{2}(x),\ldots f_{n}(x)[/latex] for which there are no constants, such that [latex]c_{1},c_{2},\ldots c_{n}[/latex], such that [latex]c_{1}f_{1}(x) + c_{2}f_{2}(x) + {\cdots}+c_{n}f_{n}(x) = 0[/latex] for all [latex]x[/latex] in the interval of interest

nonhomogeneous linear equation: a second-order differential equation that can be written in the form [latex]a_{2}(x)y^{\prime\prime}+a_{1}(x)y^{\prime}+a_{0}(x)y=r(x)[/latex] but [latex]r(x)\ne 0[/latex] for some value of [latex]x[/latex]