Y'=(y ln(y)+(yx^2))/-((x)+(2y^2)) ,y(3)=1 Find the solution of the following initial value problem.

This is a differential equation problem. Here are the steps to solve it:

1. Identify the type of differential equation: The given equation is a first order non-linear differential equation.

2. Separate the variables: We can rewrite the equation as follows to separate the variables:

   dy/dx = (y ln(y)+(yx^2)) / (-(x)-(2y^2))

   This can be rewritten as:

   (-(x)-(2y^2)) dy = (y ln(y)+(yx^2)) dx

3. Integrate both sides: Now we can integrate both sides of the equation with respect to their variables. However, the equation is non-linear and does not have a standard form, so it's not possible to solve it using elementary functions.

4. Apply the initial condition: The initial condition given is y(3)=1. This means when x=3, y=1. However, without a general solution from step 3, we can't apply the initial condition to find a particular solution.

In conclusion, this differential equation cannot be solved using standard methods due to its non-linearity and complexity. Advanced methods or numerical solutions might be needed.