Question

Solve the given initial value problem and determine at least approximately where the solution is valid. (2x - y) + (2y - x)y' = 0, y(1) = 3

Answer 1

The initial value problem (2x - y) + (2y - x)y' = 0 with y(1) = 3 involves manipulation to become separable, followed by integration and application of initial conditions to find the solution's validity near the point x = 1.

Step-by-step explanation:

Solving the Initial Value Problem

To solve the given initial value problem (2x - y) + (2y - x)y' = 0 with the initial condition y(1) = 3, we must first separate variables and integrate. This differential equation is not presented in a standard form, so we need to manipulate it to make it separable.

Step-by-step Solution

1. Rewrite the equation as (2y - x)y' = y - 2x.
2. Make y the subject in the terms involving y' to get y' = (y - 2x) / (2y - x)
3. Separate the variables so all y terms are on one side and all x terms are on the other side to integrate: ∗(2y - x) / (y - 2x)dy = ∗dx
4. Integrate both sides, apply initial conditions, and solve for the implicit solution.
5. Determine the validity of the solution based on the range of the natural logarithm function that arises from the integration and the initial conditions provided.

The solution's validity will be around the domain where the natural logarithm is defined, considering also the initial condition y(1) = 3, meaning that the solution is valid near x = 1.