Final answer:
To find the function of f(x) for the given equation, 625y" - 9y = 0, we can differentiate the equation and solve the resulting differential equation. The general solution is f(x) = Ae^(3x/25) + Be^(-3x/25).
Step-by-step explanation:
To find the function of f(x) for the equation 625y" - 9y = 0, we can assume that y is a function of x, so y = f(x). We can differentiate this equation twice using the power rule for differentiation. The first derivative of y with respect to x is y' = f'(x) and the second derivative is y" = f''(x). Substituting these derivatives into the given equation, we get 625f''(x) - 9f(x) = 0.
This is a second-order linear homogeneous differential equation. To solve this equation, we can assume a solution of the form f(x) = e^(rx), where r is a constant. Substituting this into the equation, we get 625r^2e^(rx) - 9e^(rx) = 0. Simplifying and dividing by e^(rx), we get 625r^2 - 9 = 0. Solving this quadratic equation for r, we find two solutions: r = 3/25 and r = -3/25.
Therefore, the general solution to the differential equation is f(x) = Ae^(3x/25) + Be^(-3x/25). The constants A and B can be determined using any initial conditions or boundary values, if provided.