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प्रश्न
For the following differential equation find the particular solution satisfying the given condition:
`cos("dy"/"dx") = "a", "a" ∈ "R", "y"(0) = 2`
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उत्तर
`cos("dy"/"dx") = "a"`
∴ `"dy"/"dx" = cos^-1 "a"`
∴ dy = (cos-1 a) dx
Integrating both sides, we get
`int "dy" = (cos^-1 "a") int "dx"`
∴ y = (cos-1 a) x + c
∴ y = x cos-1 a + c
This is a general solution.
Now, y(0) = 2, i.e. y = 2, when x = 0
∴ 2 = 0 + c
∴ c = 2
∴ the particular solution is
y = x cos-1 a + 2
∴ y - 2 = x cos-1 a
∴ `("y" - 2)/"x" = cos^-1 "a"`
∴ `cos (("y - 2")/"x")` = a.
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