91Ó°ÊÓ

The derivatives of a function \(y = f(x)\) of a quantity x is a measurement of the rate at which the function's value y changes when the variable x varies. The derivatives of

f with respect to x is what it's called.

Let the given function be\(y'' - 5y' + 6y = 0\).

Then, the first derivative of the function is \(y' = m{e^{mx}}\).

The second derivative of the function is \(y'' = {m^2}{e^{mx}}\).

Substitute\(y\),\(y'\)and \(y''\) into the differential equation.

\(\begin{aligned}{c}{m^2}{e^{mx}} - 5m{e^{mx}} + 6{e^{mx}} = 0\\{e^{mx}}({m^2} - 5m + 6) = 0\end{aligned}\)

As \({e^{mx}}\) will never equal to \(0\), that implies that \({m^2} - 5m + 6\).

\(\begin{aligned}{c}{m^2} - 5m + 6 = 0\\(m - 2)(m - 3) = 0\\m = 2,3\end{aligned}\)

Thus, the value of \(m\) is \(2\) or \(3\).