section%204.6

section%204.6 - Chapter 4. Higher-Order Differential...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

Ask a homework question - tutors are online