Evaluate the value of the arbitrary constant ' c ' given that y (0)=0 , of the differential equation d y d x + x y x^2-1 = x^6+4 x 1-x^2

Step-by-step solution

The given differential equation is a linear differential equation of the form dy dx + P(x)y = Q(x) , where P(x) = x x^2-1 and Q(x) = x^6+4x 1-x^2 . The integrating factor IF is given by e^ P(x) dx = e^ x x^2-1 dx = e^ 1 2 |x^2-1| = |x^2-1| . Since the domain involves 1-x^2 , we consider |x| Multiplying the differential equation by IF , we get d dx (y 1-x^2 ) = x^6+4x 1-x^2 1-x^2 = x^6+4x . Integrating both sides with respect to x , we have y 1-x^2 = (x^6+4x) dx = x^7 7 + 2x^2 + c . Given the initial condition y(0)