Final answer:
To solve the given initial-value problem dy/dx = x^2 - 2xy, y(1) = 1/3, we use the method of separable variables. By separating the variables, integrating, applying the initial condition, and simplifying the equation, we find the solution y = |x^2 - 2xy|.
Step-by-step explanation:
To solve the given initial-value problem, we need to find the function y(x) that satisfies the differential equation and the initial condition. The differential equation is dy/dx = x^2 - 2xy, and the initial condition is y(1) = 1/3. To solve this, we'll use the method of separable variables.
1. Rewrite the equation as dy/dx + 2xy = x^2.
2. Rearrange the terms to get dy/dx = x^2 - 2xy.
3. Separate the variables by dividing both sides by (x^2 - 2xy) to get (1/y)dy = (x/(x^2 - 2xy))dx.
4. Integrate both sides with respect to their respective variables. On the left side, the integral of (1/y)dy is ln|y| + C1, where C1 is the constant of integration. On the right side, the integral of (x/(x^2 - 2xy))dx requires the substitution u = x^2 - 2xy.
5. Evaluate the integral on the right side using the substitution. After integrating, we find that ln|x^2 - 2xy| + C2, where C2 is another constant of integration.
6. Combine the two sides to get ln|y| + C1 = ln|x^2 - 2xy| + C2.
7. Simplify the equation by moving the constants to one side and taking the exponential of both sides. This gives us y = e^(C2 - C1) * |x^2 - 2xy|.
8. Apply the initial condition y(1) = 1/3 to find the value of the constant e^(C2 - C1). Substitute x = 1 and y = 1/3 into the equation to get 1/3 = e^(C2 - C1) * |1^2 - 2(1)(1/3)|.
9. Solve for e^(C2 - C1) by dividing both sides by |1 - 2/3| and taking the logarithm of both sides. This gives us e^(C2 - C1) = 1/3 * 3/1 = 1.
10. Plug the value of e^(C2 - C1) = 1 back into the equation y = e^(C2 - C1) * |x^2 - 2xy| to get y = |x^2 - 2xy|.