Give the general solution to dydx 4 x sec yx yx sec yx ran

Give the general solution to dy/dx = 4 x sec (y/x) + y/x sec (y/x) ran (y/x) - 4 1n (x) = c sin (y/x) + 4 1n (x) = C sin (y/x) - 4 1n (x) = C sec (y/x) + 4 1n (x) = C sin (y/x) - 4/x^2 = C None of the above.

Solution

dy/dx = [4*x*sec (y/x) + y]/x

dy/dx = 4*sec (y/x) + y/x

Now put y/x = v

y = vx

dy/dx = x*dv/dx + v

v + x*dv/dx = 4*sec v + v

x*dv/dx = 4*sec v

dv/(sec v) = 4*dx/x

dv/sec v = sec v*tan v

dx/x = ln x

sec v*tan v = 4*ln x + C

put v = y/x

sec (y/x)*tan (y/x) - 4*ln x = C

Option A is the answer.