Final answer:
To solve the given differential equation (x² - xy)dx - x² dy = 0 using the method of integrating factor, we find the integrating factor, multiply both sides by it, rearrange the terms, integrate both sides, and solve for y.
Step-by-step explanation:
To solve the given differential equation using the method of integrating factor, we first need to rearrange it to the standard form. So we rewrite it as (x² - xy)dx - x² dy = 0. Next, we find the integrating factor, which is given by the formula μ(x) = e^(∫[P(x)]dx), where P(x) is the coefficient of dx. In this case, P(x) = -x².
So, μ(x) = e^(∫[-x²]dx) = e^(-1/3x³) = (1/x³)^(1/3). We multiply both sides of the differential equation by the integrating factor μ(x) = (1/x³)^(1/3) and rearrange the terms to obtain ((x² - xy) / x³) dx - (x² / x³) dy = 0.
Simplifying the equation further, we get (x⁻¹ - yx⁻²)dx - x⁻¹ dy = 0. Now, we can see that the left-hand side is the derivative of (yx⁻¹). Therefore, integrating both sides, we have ∫(yx⁻¹) dx - ∫x⁻¹ dy = C, where C is the constant of integration.
Finally, solving the integrals, we get yx⁻¹ - ln|x| + C = 0.