Question

For the following differential equation find the particular solution satisfying the given condition:

`cos("dy"/"dx") = "a", "a" ∈ "R", "y"(0) = 2`

Answer 1

`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.