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Unformatted text preview: Chapter 4. Higher-Order Diﬀerential Equations §4.6 Variation of Parameters We study linear nonhomogeneous DEs in the form of y + P (x)y + Q(x)y = f (x) where P (x), Q(x) and f (x) are continuous functions. The general solution of the DE is y = yc + yp where yp is a particular solution and yc is the general solution of the corresponding homogeneous DE and y c = c1 y1 + c2 y2 where y1 and y2 are two linearly independent solutions to homogeneous DE. We seek a particular solution yp of the form yp = u1 (x)y1 (x) + u2 (x)y2 (x). We can get
y f (x) W1 2 u1 (x) = =− , W W where W2 y1 f (x) u2 (x) = = W W
W= y1 y1 y2 , y2 W1 = 0 f y2 , y2 W2 = y1 0 y1 f Ex: Find the general solution. y − 4y + 4y = (x + 1)e2x y + y = cos2 x 1 ...
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