What is the general solution of the differential equation given by x^2y – xy + 4?

To find the general solution of the differential equation given by x²y – xy + 4 = 0, we can start by reorganizing the equation.

The equation can be rewritten as:

x²y – xy = -4

Next, we can factor out a common term from the left-hand side:

y(x² – x) = -4

Now, solving for y gives us:

y = rac{-4}{x² – x}

This expression for y describes the relationship between y and x for our equation. However, we are also interested in finding a more general solution that includes all potential solutions.

We will analyze the denominator: x² – x = x(x – 1). This indicates that our function will have undefined points when x = 0 and x = 1. Consequently, we should consider these critical points when constructing our solution.

The general solution can be expressed as:

y = rac{-4}{x(x – 1)} + c

Here, c is a constant that can be determined through initial or boundary conditions if they are provided. This general solution includes the particular solution for any values of x, as long as x is not equal to 0 or 1.

In conclusion, the general solution of the differential equation x²y – xy + 4 = 0 is:

y = rac{-4}{x(x – 1)} + c