Use the separation of variables method to find the solution of the first-order separable differ-ential equationyx y′ = y2 + 9x2 + 4which satisfies y(0) = 3

The given differential equation is yx y′ = y^2 + 9x^2 + 4.

First, we rewrite the equation in the form of dy/dx = (y^2 + 9x^2 + 4)/(yx).

Next, we separate the variables by moving the y-related terms to one side and the x-related terms to the other side of the equation. This gives us (yx)/(y^2 + 9x^2 + 4) dx = dy.

Now, we integrate both sides of the equation.

∫(yx)/(y^2 + 9x^2 + 4) dx = ∫dy.

Unfortunately, the integral on the left side is not straightforward to solve. The equation is not easily separable and cannot be solved using standard methods of integration.

Therefore, it seems there might be a mistake in the problem as it is not a standard separable differential equation. Please check the problem and try again.