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Unformatted text preview: Linear Eqs With Constant Coefficients We still want to consider 2nd order linear equations of the form y 00 + p ( x ) y + q ( x ) y = 0 but now we will restrict our attention to the case where p ( x ) and q ( x ) are constant functions. We wont lose anything by allowing a nonzero constant coefficient of y 00 as well, since we could always divide it back out. Thus we will consider equations of the form ay 00 + by + cy = 0 . From our previous theorem, we are guaranteed that this equation has a general solution of the form y ( x ) = c 1 y 1 ( x ) + c 2 y 2 ( x ) , provided that we can find two linearly independent solutions y 1 and y 2 (i.e. solutions whose Wronskian is nonzero). An Educated Guess Regarding the solutions y 1 and y 2 , we are left with no alternative but to try to make an educated guess. From that the equation ay 00 + by + cy = 0 we can make a few observations: (1) The functions y,y , and y 00 must be of a similar shape, since constant multiples of them add to zero. For example no multiple ofto zero....
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