> [!NOTE] Theorem (homogeneous second order linear equations with constant coefficients) > Let $a \frac{d^{2}}{dt^{2}}x(t) +b \frac{d}{dt} x(t) + cx(t) = 0 \tag{1}$be a [[Homogeneous Ordinary Differential Equation|homogenous]] [[Second Order Linear Scalar Ordinary Differential Equation|second order linear ordinary differential equation]] with real coefficients $a\neq 0,b,c.$ > > Let $\lambda_{1,2}$ be the roots of the auxiliary equation: $a\lambda^{2}+b\lambda+c=0$ which is given by $\lambda_{1,2}=-\frac{b}{2a} \pm \frac{1}{2a} \sqrt{ b^{2}-4ac }.$Then all [[Solution to Scalar Ordinary Differential Equation|solutions]] to $(1)$ are of the form $x(t) = \begin{cases} l_{1}e^{\lambda_{1}t} + l_{2} e^{\lambda_{2}t} & b^{2}-4ac > 0, \\ l_{1}e^{\lambda t} + l_{2}te^{\lambda t} & b^{2}-4ac=0 \;(\text{where }\lambda:= \lambda_{1}=\lambda_{2}), \\ e^{p t} (l_1 \sin q t + l_2 \cos q t) & b^{2}-4ac<0 \; (\text{where } \lambda_{1,2}=p \pm i\,q ) \end{cases}$for some $l_{1},l_{2}\in \mathbb{R}.$ **Proof**: ... Follows from [[Solution Space of Homogenous Second Order Linear Ordinary Differential Equation is Two-dimensional]] and ... # Applications **Solution to non-homogenous second order equations**: [[Solution to Inhomogeneous Second Order Linear Scalar Ordinary Differential Equation with Real Coefficients]] asserts that...