The solution of differential equation
` (dt)/(dx) = (t[d/dx{g(x)}]-t^(2))/(g(x) )` is

Options

A. `t=(g(x)+C)/x`
B. ` t = g(x)/x+C`
C. ` t = (g(x))/(x+C) `
D. ` t = g(x) + x+C`

Answer

c

Explanation

Given ` (dt)/(dx) -t (g'(x))/(g(x)) = - (t^(2))/(g(x))`
` rArr -1/(t^(2)) (dt)/(dx) + (1g'(x))/(tg(x))+1/(g(x))" " ` …(i)
Let ` z = 1/t rArr 1/(t^(2)) (dt)/(dx) = (dz)/(dx)`
` :. `From Eq. (i) ` (dz)/(dx) +(g'(x))/(g(x)) z = 1/(g(x)`
On comparing with `(dz)/(dx) +Pz = Q ` , we get
` P = (g'(x))/(g(x)),Q = 1/(g(x))`
` :. IF = e^(int(g'(x))/(g(x))dx)= e^(log [g(x)])=g(x)`
Ths , its general solution is
` z * g(x) = int g(x) * 1/(g(x)) dx + C`
` rArr 1/t g(x) = x+C rArr (g(x))/(x+C)=t`