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Question
Differential equation representing the family of curves y = ex (Acosx + Bsinx) is `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + 2y` = 0
Options
True
False
Solution
This statement is True.
Explanation:
Given equation is y = ex (Acosx + Bsinx)
Differentiating both sides, we get
`("d"y)/("d"x)` = ex (–A sin x + B cos x) + (A cos x + B sin x) ex
`("d"y)/("d"x)` = ex (–A sin x + B cos x) + y
Again differentiating w.r.t. x, we get
`("d"^2y)/("d"x^2) = "e"^x (-"A" cosx - "B" sinx) + (-"A" sinx + "B"cosx) . "e"^x + ("d"y)/("d"x)`
`("d"^2y)/("d"x^2) = "e"^x ("A" cos x + "B" sin x) + ("d"y)/("d"x) - y + ("d"y)/("d"x)`
`("d"^2y)/("d"x^2) = - y + y + 2("d"y)/("d"x)`
∴ `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + 2y` = 0
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