Final answer:
To solve the differential equation dy/dx = xy^2, we can use an appropriate substitution. By substituting u = y^(-1), we can transform the equation into a separable differential equation and solve for y.
Step-by-step explanation:
To solve the differential equation dy/dx = xy^2, we can use an appropriate substitution. Let's substitute u = y^(-1). Differentiating with respect to x, we have du/dx = -1/y^2 * dy/dx. Now we substitute this in the given equation, so we have -du/dx = x/y^2. Rearranging the terms, we get du/dx = -xu.
This is now a separable differential equation. We can rewrite it as du/u = -xdx. Integrating both sides, we get ln|u| = -x^2/2 + C, where C is the constant of integration. Exponentiating both sides, we have |u| = e^(-x^2/2 + C). Since we are only interested in the positive values of u, we can drop the absolute value sign and write u = e^(C) * e^(-x^2/2).
Substituting back u = y^(-1), we have y = e^(-C) * e^(x^2/2), where e^(-C) is a constant. Therefore, the solution to the differential equation is y = Ae^(x^2/2), where A is a constant.