Differential EquationHard
Question
If y = e-x cos x and yn + kn y = 0, where yn = 
and kn, n ∈ N are constants.
and kn, n ∈ N are constants.Options
A.k4 = 4
B.k8 = - 16
C.k12 = 20
D.k16 = - 24
Solution
y = e-x cos x
y1 = -e-x cos x - e-x sin x = - e-x (sin x + cosx)
= - √2 e-x (cos x cos
+ sinx sin 
)
y1 = - √2 e-x cos
y2 = + √2 e-x cos
+ √2 e-x sin 
= √2 e-x
= √2 . √2 e-x cos
similarly
= (√2)2 e-x cos
y3 = - (√2)3 e-x cos
y4 = + (√2)4 e-x cos
= - (Ω2)4 e-x cosx
y4 + 4 y = 0
Similarly
y8 = (√2)8 e-x cos
= (√2)3 e-x cos (x - 2π)
y1 = -e-x cos x - e-x sin x = - e-x (sin x + cosx)
= - √2 e-x (cos x cos
+ sinx sin )y1 = - √2 e-x cos
y2 = + √2 e-x cos
+ √2 e-x sin = √2 e-x
= √2 . √2 e-x cos
similarly
= (√2)2 e-x cos
y3 = - (√2)3 e-x cos
y4 = + (√2)4 e-x cos
= - (Ω2)4 e-x cosx
y4 + 4 y = 0
Similarly
y8 = (√2)8 e-x cos
= (√2)3 e-x cos (x - 2π)Create a free account to view solution
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