Greens theorem and the variation of parameters

To solve a second-order inhomogeneous ordinary differential equation, either the Green s function method or the variation of parameters method can be used. Consider the self-adjoint equation 

Following the method outlined in Sect. 3, above, the solution y(xQ) is 

The variation of parameters method uses the two linearly independent solutions of the homogeneous equation (332), y y (x) and y2 (x) (which, of course, also appear in the Green s function) and so 

Clearly, W(yj, y2) is zero if y L = constant xy2. If y2 andy3 are linearly independent, then the Wronskian is non-zero for some values of sc. In addition, if we multiply eqn. (334a) by y, eqn. (334c) by yl7 subtract and integrate by parts, we have 

A similar equation may be written with y2 in place of yi. Multiply the former by y2 and the latter equation by y1, subtract to eliminate y and so 

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