The general solution of d y d x = ⁻¹ x is

Step-by-step solution

The given differential equation is dy dx = ⁻¹ x . Integrating both sides with respect to x , we get y = ⁻¹ x , dx . Using integration by parts, let u = ⁻¹ x and dv = dx . Then du = 1 1-x^2 dx and v = x . The formula for integration by parts is u , dv = uv - v , du . Substituting the values, we get y = x ⁻¹ x - x 1-x^2 dx . To evaluate the integral x 1-x^2 dx , let t = 1 - x^2 . Then dt = -2x , dx , which implies x , dx = - 1 2 dt . Substituting these into the integral, we get x 1-x^2 dx = -1/2 t dt = - 1 2 t^ -1/2