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For the following differential equation find the particular solution satisfying the given condition: dydxaaRycos(dydx)=a,a∈R,y(0)=2 - Mathematics and Statistics

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Sum

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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Solution

`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

∴ `("y" - 2)/"x" = cos^-1 "a"`

∴ `cos (("y - 2")/"x")` = a.

Concept: Formation of Differential Equations
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