Final answer:
To solve the differential equation x²y'' - xy' + x = 0, we first need to convert it to standard form by dividing by x². The standard form doesn't require an integrating factor as it lacks a y term. Solutions can be sought by substituting y = xm into the standard form and solving the resulting characteristic equation.
Step-by-step explanation:
The differential equation given is x²y'' - xy' + x = 0. To find its general solution, we first need to transform it into standard form. This is done by dividing the entire equation by x², assuming x ≠ 0, obtaining y'' - (1/x)y' + (1/x²) = 0, which is a second-order linear homogeneous differential equation. To find the integrating factor μ(x) for such an equation, we need it to be in the form y'' + p(x)y' + q(x)y = 0. However, since there is no y term in the original equation, the concept of an integrating factor doesn't apply directly here as it typically would for a first-order equation.
Instead, we proceed to solve the standard form equation by looking for solutions of the form y = xm. We substitute y into the standard form equation to obtain a characteristic equation in terms of m. Solving this will give us the exponents necessary to express the general solution of the differential equation.