The differential equation given in the image is:

$(x^2 + y^2) \, dx + 3xy \, dy = 0$

This is a homogeneous differential equation. We can solve it by using the substitution $y = vx$, where $v$ is a function of $x$, and then follow the steps to simplify the equation into a separable form.

Steps to solve:

1. Substitute $y = vx$, hence $dy = v \, dx + x \, dv$.

2. Rewrite the equation: Substituting $y = vx$ and $dy = v \, dx + x \, dv$ into the original equation:

   $(x^2 + (vx)^2) dx + 3x(vx)(v dx + x dv) = 0$ Simplify and separate the variables.

3. Simplify and solve: The equation will reduce into a form that you can integrate to find the solution.

Would you like a detailed step-by-step solution for this? If yes, let me know!

Here are 5 related questions to explore:

1. What is the general form of a homogeneous differential equation?
2. How does the substitution $y = vx$ simplify homogeneous differential equations?
3. What are some examples of non-homogeneous differential equations?
4. How can you verify if a differential equation is homogeneous?
5. What are the applications of solving homogeneous differential equations in physics?

Tip: Always check if a differential equation can be simplified by substitution, especially for homogeneous equations, as it helps in separating variables easily!