MATH201 Chapter Notes - Chapter 3: Wronskian, Linear Map, Damping Ratio
16 Apr 2014
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Consider the equation x2y + xy y = 0. By inspection, 1(x) = x is a solution. The equation becomes xv + 3v = 0. 1 x2 6= constant so that 1 and 2 are linearly independent. The general solution to the original de is y(x) = c1 x + c2 x. Another concept that will prove to be useful in characterizing solutions to linear di erential equations is that of the wronskian. The wronskian of two functions f and g is de ned as f (x) f (x) 1 2) q(x) ( 1 2 1 2) = p (x)w. Hence w = ae r p (x) dx. If a = 0, then w 0, and if a 6= 0, then w 6= 0 for all x (a, b). The following result gives a nice characterization of the solutions of linear, homogeneous di erential equations. Consider the nonhomogeneous equation y + p (x)y + q(x)y = g(x). (3. 12)
