- #1
bobmerhebi
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Homework Statement
solve:
y" + 4y' + 4y = (3 + x)e-2x ... (1); y(0) = 2 & y'(0) = 5
Homework Equations
using Undetermined Coefficients Method
The Attempt at a Solution
solving the associated homog. eq of (1) : y" + 4y' + 4y = 0 ... (2)
we get: yc = c1 e-2x as m=-2
let f(x) = (3 + x)e-2x
the UC set of :
a) 3e-2x = {e-2x} = S1
b) xe-2x = {xe-2x , e-2x} = S2
as S1 is included in S2 then we omit S1
but e-2x is a solution of (2) as its in yc then we multiply the 2ns set by x to get:
S'2 = {x2 e-2x, xe-2x } where both of the elements are NOT sol.'s of (2)
hence we get yp = Axe-2x + Bx2 e-2x
y'p = Ae-2x -2Axe-2x + 2Bxe-2x - 2Bx2 e-2x
& y"p = -4Ae-2x + 4Axe-2x + 2Be-2x - 8Bxe-2x +4Bx2 e-2x
substitution in (1) one gets:
2Be-2x = (3+x)e-2x so B = 3+x/2 ? ??
where is A? why is B interms of x?
please help
thx