Rewrite the Bernoulli differential equation in standard form.

You're on a roll — Step 1: Rewrite the Bernoulli differential equation in standard form. The given equation is (dy)/(dx) - (y)/(x) = y^2 (x). This is a Bernoulli equation of the form (dy)/(dx) + P(x)y = Q(x)y^n, where P(x) = -(1)/(x), Q(x) = (x), and n=2. Step 2: Transform the Bernoulli equation into a linear first-order differential equation. Divide the entire equation by y^2: y^-2(dy)/(dx) - (1)/(x)y^-1 = (x) Let v = y^1-n = y^1-2 = y^-1. Differentiate v with respect to x: (dv)/(dx) = -1 · y^-2 (dy)/(dx) So, y^-2(dy)/(dx) = -(dv)/(dx). Substitute v and (dv)/(dx) into the transformed equation: -(dv)/(dx) - (1)/(x)v = (x) Multiply by -1 to get the standard linear form (dv)/(dx) + P(x)v = Q(x): (dv)/(dx) + (1)/(x)v = -(x) Step 3: Find the integrating factor. For this linear equation, P(x) = (1)/(x). The integrating factor I(x) is: I(x) = e^ P(x) dx = e^ (1)/(x) dx = e^|x| = x (We assume x > 0 for simplicity). Step 4: Multiply the linear equation by the integrating factor and integrate. Multiply (dv)/(dx) + (1)/(x)v = -(x) by x: x(dv)/(dx) + v = -x(x) The left side is the derivative of the product (xv): (d)/(dx)(xv) = -x(x) Integrate both sides with respect to x: (d)/(dx)(xv) dx = -x(x) dx xv = - x(x) dx To evaluate x(x) dx, use integration by parts ( u\,dv = uv - v\,du). Let u=x and dv=(x)dx. Then du=dx and v=(x). x(x) dx = x(x) - (x) dx = x(x) - (-(x)) + C_1 = x(x) + (x) + C_1 So, xv = -(x(x) + (x)) + C xv = -x(x) - (x) + C Step 5: Substitute back v = y^-1 and solve for y. x((1)/(y)) = -x(x) - (x) + C (x)/(y) = C - x(x) - (x) Solve for y: y = (x)/(C - x(x) - (x)) The solution to the Bernoulli differential equation is: y = (x)/(C - x(x) - (x)) What's next?