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Question
Which of the following is the general solution of `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + y` = 0?
Options
y = (Ax + B)ex
y = (Ax + B)e–x
y = Aex + Be–x
y = Acosx + Bsinx
Solution
y = (Ax + B)ex
Explanation:
The given differential equation is `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + y` = 0
Since the above equation is of second order and first degree
∴ `"D"^2y - 2"D"y + y` = 0
Where D = `"d"/("d"x)`
⇒ `("D"^2 - 2"D" + 1)y` = 0
∴ Auxiliary equation is m2 – 2m + 1 = 0
⇒ (m – 1)2 = 0
⇒ m = 1, 1
If the roots of Auxiliary equation are real and equal say (m)
Then CF = `("c"_1 + "c"_2) . "e"^(mx)`
∴ CF = `("A"x + "B")"e"^x`
So y = `("A"x + "B")."e"^x`
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