Find an explicit solution of the given initial value problem: x² dy/dx + y = xy + 1/4

To solve the initial value problem given by the equation x² dy/dx + y = xy + 1/4, we start by rearranging it into a more manageable form.

First, we can rewrite the differential equation as follows:

x² dy/dx = xy + 1/4 - y

This simplifies to:

dy/dx = (xy + 1/4 - y)/x²

Next, we’ll reorganize the right side:

dy/dx = (y(x - 1) + 1/4)/x²

This is a first-order linear ordinary differential equation. We can use an integrating factor to solve it.

The integrating factor, μ(x), for a linear equation of the form dy/dx + P(x)y = Q(x) is given by:

μ(x) = e^(∫P(x)dx)

Here, our P(x) is -1/x², so:

μ(x) = e^(∫(-1/x²)dx) = e^(1/x)

Multiplying through the original equation by the integrating factor:

e^(1/x) dy/dx + e^(1/x)(-1/x²)y = e^(1/x)(1/4)/x²

Now, the left-hand side can be expressed as the derivative of a product:

d(e^(1/x)y)/dx = e^(1/x)(1/4)/x²

Integrating both sides with respect to x gives:

e^(1/x)y = ∫(e^(1/x)(1/4)/x²)dx

This integral can be solved using integration techniques. After performing the integral and solving for y, we would obtain:

y = (1/4)e^(-1/x)(x + C)

Now, using any initial condition you may have from your problem (which isn’t provided), you would be able to find the value of the constant C.

In conclusion, the explicit solution for the problem will generally take the form shown above, adjusted based on the initial value provided:

y = (1/4)e^(-1/x)(x + C)