Let y = y(x) be the solution of the differential equation dy dx + 2xy = 2x^3 + 2x with the initial condition y(0) = . If the solution curve y(x) has exactly one local minimum and no local maximum on R , then the maximum possible integral value of is

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) = 2x and Q(x) = 2x^3 + 2x . The integrating factor (I.F.) is: I.F. = e^ 2x dx = e^ x^2 Multiplying the equation by the I.F. and integrating: y e^ x^2 = (2x^3 + 2x)e^ x^2 dx + C To evaluate the integral, let t = x^2 , so dt = 2x dx . The integral becomes: (t + 1)e^t dt Using integration by parts, this evaluates to (t + 1)e^t - e^t = te^t . Substituting back t = x^2 , we get: (2x^3 + 2x)e^ x^2 dx = x^2 e^ x