Differential EquationMedium
Question
Number of values of m ∈ N for which y = emx is a solution of the differential equation D3y -3D2y - 4Dy +12y = 0 is -
Options
A.0
B.1
C.2
D.more than 2
Solution
y = emx then D(y) = memx ,D2 (y) = m2emx
D3 (y) = m3 emx
then given
D3 y - 3D2y - 4Dy + 12y = 0
⇒ m3 emx - 3m2emx - 4memx +12emx = 0
⇒ m3 - 3m2 - 4m + 12 = 0 (∵ emx ≠ 0)
⇒ (m - 2) (m - 3) (m + 2) = 0
⇒ m = 2, 3, - 2
Hence number of values of m ∈ N will be 2.
D3 (y) = m3 emx
then given
D3 y - 3D2y - 4Dy + 12y = 0
⇒ m3 emx - 3m2emx - 4memx +12emx = 0
⇒ m3 - 3m2 - 4m + 12 = 0 (∵ emx ≠ 0)
⇒ (m - 2) (m - 3) (m + 2) = 0
⇒ m = 2, 3, - 2
Hence number of values of m ∈ N will be 2.
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