The solution of the differential equation x^2 dy dx - xy = 1 is...

Step-by-step solution

The given differential equation is x^2 dy dx - xy = 1 Dividing by x^2 , we get: dy dx - 1 x y = 1 x^2 This is a linear differential equation of the form dy dx + Py = Q , where P = - 1 x and Q = 1 x^2 Integrating Factor (IF) = e^ P dx = e^ - 1 x dx = e^ - x = 1 x The solution is given by: y ( IF ) = Q ( IF ) dx + c y ( 1 x ) = 1 x^2 ( 1 x ) dx + c y x = x⁻³ dx + c y x = x⁻² -2 + c y x = - 1 2x^2 + c Multiplying the entire equation by 2x^2 , we get: 2xy = -1 + 2cx^2 2xy - 2cx^2 + 1 = 0 Answer: 2xy - 2cx^2 + 1 = 0