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.

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

First, we rewrite the equation in the form of dy/dx = f(x,y).

So, yx dy/dx = y^2 + 9x^2 + 4, which simplifies to dy/dx = (y^2 + 9x^2 + 4)/yx.

Now, we separate the variables.

This gives us (yx)/(y^2 + 9x^2 + 4) dx = dy.

Integrating both sides, we get ∫(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 it seems there may be a mistake in the problem.

If the equation was yx dy/dx = y^2 - 9x^2, for example, we could separate variables and solve. But as it stands, the equation yx dy/dx = y^2 + 9x^2 + 4 does not appear to be solvable using standard methods of separation of variables.