The equation (dy)/(dx) = A(x)y² + B(x)y + C(x) is called a Riccati equation. Suppose that one particular solution y1(x) of this equation is known. Then the substitution y=y1 + 1/v transforms the Riccati equation into the linear equation dv/dx + (B + 2Ay1) v = -A.
Use the method of Problem 63 to solve the equations in Problems 65, given that y_1(x) = x is a solution of each.
dy/dx + 2xy = 1 + x² + y²