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A second-order linear differential equation involves a function and its derivatives up to the second order.
This type of equation is a vital concept in mathematics, especially because it models various physical phenomena such as oscillations and waves.
Let's break down the terminology:

• "Second-order" refers to the highest derivative being the second derivative (\(z''\)). It indicates how the function evolves with second-degree changes.
• "Linear" means the equation is linear in the unknown function and its derivatives. For example, the given differential equation, \(9 z^{\prime \prime}-z=0\), is linear. There are no powers or products of \(z, z', z''\).

The form of a second-order linear differential equation can be represented as:\[a z'' + b z' + c z = 0\]If the equation consists solely of constant coefficients, like in the example \(a = 9, b = 0, c = -1\), the analysis becomes straightforward. Constant coefficients mean that the coefficients of the derivatives do not vary with respect to the independent variable, which simplifies solving the equation.

The characteristic equation is an auxiliary equation obtained by substituting a meaningful form of the solution into the differential equation.
This equation is crucial because it simplifies solving the differential equation.
For a given second-order linear differential equation:\[a z'' + b z' + c z = 0\]We replace the derivatives as follows:

• \(z'' \rightarrow r^2\)
• \(z' \rightarrow r\)
• \(z \rightarrow 1\)

Doing so transforms our differential equation into the characteristic equation:\[9r^2 - 1 = 0\]Solving this equation keeps the process orderly and helps us find the roots, essential for determining the general solution.
In this context, the quadratic nature (\(r^2\)) reflects the second-order character, and the real roots (\(r = \pm\frac{1}{3}\)) hint at exponential growth or decay, leading us to the solution's structure.

A linear homogeneous differential equation is one where every term is either the unknown function or its derivatives.
Such an equation doesn't include terms independent of the unknown function (i.e. there are no standalone terms on the right side of the equation).
Considering our example:\[9 z^{\prime \prime}-z=0\]"Homogeneous" implies that if we substitute the zero function for the unknown function, the equation holds true.
This property tells us that the symmetries in the differential equation form solutions that can be superimposed or combined, reflecting the general solution's structure. The general solution to such an equation involves combining all possible solutions, which are expressible and solvable using the roots of the characteristic equation.
In this context, a linear homogeneous differential equation with constant coefficients yields solutions like:\[z(t) = C_1 e^{r_1 t} + C_2 e^{r_2 t}\]where the roots (\(r_1, r_2\)) are related directly to the characteristic equation, \(r_1 = \frac{1}{3}\) and\(r_2 = -\frac{1}{3}\) in our problem.