The integrating factor of the differential equation (1 + t^2) + (x - e^ ⁻¹t ) dt dx = 0 is

Step-by-step solution

The given differential equation is (1 + t^2) + (x - e^ ⁻¹t ) dt dx = 0 Rearranging the terms, we get: (1 + t^2) dx dt + x - e^ ⁻¹t = 0 dx dt + 1 1 + t^2 x = e^ ⁻¹t 1 + t^2 This is a linear differential equation of the form dx dt + Px = Q , where P = 1 1 + t^2 and Q = e^ ⁻¹t 1 + t^2 The integrating factor is given by e^ P dt IF = e^ 1 1 + t^2 dt = e^ ⁻¹t Answer: e^ ⁻¹t