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 $y_{1}(x)$ and $y_{2}(x)$ are linearly independent solutions to a second-order, linear, homogeneous differential equation, then the general solution is given by $y(x)=c_{1}y_{1}(x)+c_{2}y_{2}(x)$ .
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
$a_{2}(x)y^{\prime\prime}+a_{1}(x)y^{\prime}+a_{0}(x)y=r(x)$
Second-order equation with constant coefficients
$ay^{\prime\prime}+by^{\prime}+cy=0$

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 $ay^{\prime\prime}+by^{\prime}+cy=0$

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

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

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

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