Final answer:
To find the explicit solution, we divide the equation by x^2, rearrange it, integrate, and solve for y using different cases. The solution is either y = e^(-1/x + C)/(1 - x) or y = -e^(-1/x + C)/(1 - x).
Step-by-step explanation:
To find an explicit solution of the initial-value problem x²(dy/dx) = y - xy, y(-1) = -3, we first divide both sides of the equation by x² to isolate the derivative of y. This gives us dy/dx = (y - xy)/x². Next, we can rearrange the equation to get dy/(y - xy) = dx/x². We can now integrate both sides of the equation with respect to their respective variables.
Integrating the left side, we get ln|y - xy| = -1/x + C, where C is the constant of integration. Next, to solve for y, we can exponentiate both sides of the equation. This gives us |y - xy| = e^(-1/x + C). Applying the absolute value, we have two cases to consider: y - xy = e^(-1/x + C) or y - xy = -e^(-1/x + C).
We can now solve each case separately. In the first case, y - xy = e^(-1/x + C), rearranging the equation gives us y(1 - x) = e^(-1/x + C). Dividing both sides by (1 - x), we find y = e^(-1/x + C)/(1 - x). In the second case, y - xy = -e^(-1/x + C), rearranging the equation gives us y(1 - x) = -e^(-1/x + C). Dividing both sides by (1 - x), we find y = -e^(-1/x + C)/(1 - x).