Final answer:
The integrating factor u(x) for the differential equation is x^{-3}, and the general solution y(x) is -x^2 + Cx^3, where C is an arbitrary constant.
Step-by-step explanation:
To find the general solution to the differential equation x² - 3xy + x\frac{dy}{dx} = 0, we need to first find the integrating factor which is a function u(x), that when multiplied with the differential equation, makes it exact. An exact equation means the left side of the equation can be expressed as the derivative of some function of x and y.
To solve the given equation, arrange it to the standard form of a linear first-order differential equation:
\frac{dy}{dx} - \frac{3}{x}y = -x
The integrating factor u(x) is found using the formula: u(x) = e^{\int -\frac{3}{x} dx} which simplifies to u(x) = x^{-3}. Multiplying the entire differential equation by the integrating factor, we get:
x^{-3}\frac{dy}{dx} - 3x^{-4}y = -1
Now, recognize that the left side of the equation is the derivative of u(x)y, and integrate both sides with respect to x to find y(x). Thus, the general solution for y(x) includes an arbitrary constant C and has the form:
y(x) = x^3\int -x^{-2} dx + Cx^3 which simplifies to y(x) = x^3(-\frac{1}{x}) + Cx^3 = -x^2 + Cx^3.