> [!NOTE] Definition (Second Order Linear Ordinary Differential Equation) > > A *linear second order ordinary differential equation* is a [[Linear Differential Equation|linear]] [[Ordinary Differential Equation|ordinary differential equation]] of the form: $a(t) \frac{d^{2}}{dt^{2}}x(t) +b(t)x(t) + c(t)x(t) = s(t), \quad x(t_{0}) = x_{0}, x'(t_{0})=v_{0} \tag{1}$with functions $a(t) \neq 0, b(t), c(t)$ and $s(t)$, $t\in (\alpha,\beta)$. # Properties **Trivial equation**: Consider the simplest second order equation $\frac{d^2}{dt^{2}}x(t)=f(t).$ We can solve it by integrating twice: $x(t)=F(t)+c_{1}t+c_{2}$ where $F$ is an antiderivative of an antiderivative of $f(t).$ **Solution space of homogenous equations**: First note that [[Solution Space of Homogenous Second Order Linear Ordinary Differential Equation Forms a Vector Space]]. Moreover [[Solution Space of Homogenous Second Order Linear Ordinary Differential Equation is Two-dimensional]]. **Solution space of non-homogenous equations:** For non-homogenous we have [[Superposition Principle for non-homogenous Second Order Linear Ordinary Differential Equation]]. # Applications **Example**: [[Second Order Linear Scalar Ordinary Differential Equation with Real Coefficients]].