Unformatted text preview: sin(2 1,
Associated homogeneous equation: y 00 y = 0.
x
2. (a) y 000 + e =ysin(2x) x +e1) ., 1 = 0. (0) = 1 1, 1. Corresponding solutions: e x , ex
y x = ln (
c) y
y Roots:
Auxiliary equation:2 2
Associated homogeneous equation: Notes, these are linearly independent on IR.
y 00 y
By Proposition 3.30 in the Course and it = 0.
This equation is of 2ﬁrst order, x
is also linear, so both of our ExAuxiliary equation:h (x) =1c= 0x Roots:, 11. Corresponding solutions: e x , ex
. + c2 e
General solution: y Theorems apply. 1,Thex theorem for linear equations
<<
1e
istence/Uniquenessin the Course Notes, these are 1.
By Proposition 3.30
RHS of inhomogeneous DE: F (x) = sin(2x) 2e x linearly) independent on IR.
= F1 (x + F2 (x).
General solution: yh (x) = c1 e x + 1 (xx , yp1 (x) = < 1. x) + M2 cos(2x).
Form of particular solution: For F c2 e): 1 < x M1 sin(2
x
RHSF2 (x): yp2 (x) = Le x . FThis=is already 2e solution(x) + F2 (x).
= F1 of the associated homogeneous
For of inhomogeneous DE: (x) sin(2x) a
Form of particular solution: yp2(xF1=xLxep1x .x) = M1 sin(2x) + M2 cos(...
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 Winter '14
 Heterogeneity, Homogeneity, yp

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