solve x(1-4y)dx - (x^2 +1)dy = 0 with y(2) = 1

This is a first order differential equation. We can solve it using the method of integrating factors.

Step 1: Rewrite the equation in standard form The standard form of a first order differential equation is dy/dx + P(x)y = Q(x). So, we rewrite the equation as:

dy/dx + [x(1-4y)]/[(x^2 +1)] = 0

Step 2: Identify P(x) and Q(x) From the standard form, we can identify P(x) = [x(1-4y)]/[(x^2 +1)] and Q(x) = 0.

Step 3: Find the integrating factor e^(∫P(x) dx) The integrating factor is e^(∫P(x) dx) = e^(∫[x(1-4y)]/[(x^2 +1)] dx).

Step 4: Multiply every term of the differential equation by the integrating factor We get e^(∫[x(1-4y)]/[(x^2 +1)] dx) * dy/dx + e^(∫[x(1-4y)]/[(x^2 +1)] dx) * [x(1-4y)]/[(x^2 +1)] = 0.

Step 5: The left-hand side of this equation is the derivative of (e^(∫P(x) dx) * y) with respect to x So, we can write the equation as d/dx [e^(∫[x(1-4y)]/[(x^2 +1)] dx) * y] = 0.

Step 6: Integrate both sides with respect to x We get e^(∫[x(1-4y)]/[(x^2 +1)] dx) * y = C, where C is the constant of integration.

Step 7: Solve for y Finally, we solve for y to get the solution of the differential equation.

Note: This is a general method to solve first order differential equations. However, this particular equation seems to be a bit complicated to solve by hand due to the presence of y in the function P(x). It might be easier to solve it using a computer algebra system.