Show that the differential equation xdy/dxsin(y/x) + x - ysin(y/x) = 0 is homogeneous. Find the particular solution of this differential equation given that x = 1 when y = π/2.

xdy/dxsin(y/x) = ysin(y/x) - x
dy/dx = y/x - sin(y/x)
Let F(x,y) = y/x - sin(y/x)
Therefore, F(λx,λy) = λy/λx - sin(λy/λx) = y/x - sin(y/x) = λ⁰F(x,y)
Given differential equation is homogeneous
Let y = vx
dy/dx = v + xdv/dx
Now, x(v + xdv/dx)sinv + x - vxsinv = 0
vxsinv + x²sinvdv/dx + x - vxsinv = 0
sinvdv = -dx/x
∫sinvdv = -∫dx/x
-cosv = -log|x| + c
-cos(y/x) = -log|x| + c
cos(y/x) = log|x| + c
When x = 1, y = π/2
cos(π/2) = log|1| + c
0 = 0 + c
=> c = 0
cos(y/x) = log|x|
x = e^(cos(y/x))
Therefore, cos(y/x) = log x