For the following differential equation find the particular solution satisfying the given condition: dydxaaRycos(dydx)=a,aR,y(0)=2

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.