find the general solution of the following differential equation by using homogenous method and separating method

Question

dy/dx = (x sec(y/x)+y) /x

Answer 1

dy/dx = sec(y/x) + y/x Let a = y/x then y = ax so, dy/dx = (x)da/dx + a

Substituting, we have

(x)da/dx + a = sec(a) + a or (x)da/dx = sec(a) then da/sec(a) = dx/x = cos(a) da

Integrating, we have sin(a) = ln(x) + constant or sin(a) = { ln(x) + ln(K) } = ln(Kx)

Therefore, a = sin^-1(ln(Kx)) then y/x = sin^-1(ln(Kx))

Finally, y = (x)sin^-1(ln(Kx)) (K = constant)