8.10.5 Differential Equation for f(x) and g(x) We will now show that the functions f(x) and g(x) satisfy the same differential equation. Taking the derivative of the first expression in equation...

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Extracted text: 8.10.5 Differential Equation for f(x) and g(x) We will now show that the functions f(x) and g(x) satisfy the same differential equation. Taking the derivative of the first expression in equation (8.489) gives d f (x) dx2 dd(x) dg(x) dx dx 1 df (x)] d(x) dx d In d(x)] df (x) ´d d(x) 2| - d(x)[d(x)f(x)] dx - (d(x)]² f(x). (8.538) dx dx Bringing all terms to the left side produces the expression d f(x) d In d(x)] df (x) dx2 + [d(x)]² f (x) = 0. dx (8.539) dx Now d(x) = d0(x)/dx = 0'(x). Therefore, the last equation can be rewritten to the form C" (2) - r(2) + (@'(x)* f (x) = 0, 0" (x) "(x) (8.540) where the prime notation is used to denote differentiation with respect to x. Exactly the same differential equation is satisfied by g(x). Given 0(x), the above differential equation is linear, of second-order, and has coefficients that are x dependent. This type of equation is known as a generalized Mathieu differential equation.