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\(2y'' + 7y' - 4y = 0\).

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.

\(\begin{aligned}{c}2{m^2}{e^{mx}} + 7m{e^{mx}} - 4{e^{mx}} = 0\\{e^{mx}}(2{m^2} + 7m - 4) = 0\end{aligned}\)

As \({e^{mx}}\) will never equal to \(0\), that implies that \(2{m^2} + 7m - 4\).

\(\begin{aligned}{c}2{m^2} + 7m - 4 = 0\\(2m - 1)(m + 4) = 0\\m = \frac{1}{2}, - 4\end{aligned}\)

Thus, the value of \(m\) is \(\frac{1}{2}\) or \( - 4\).