The given differential equation is:

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

This is a first-order differential equation, and we will try to determine its type and approach to solving it. Let's first simplify it step by step.

Step 1: Rearrange the equation

We can rewrite the equation as:

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

This sets it up in a clearer form to determine the method of solution.

Step 2: Check if it’s separable

A separable differential equation has the form:

$f(x) \, dx + g(y) \, dy = 0$

Unfortunately, this equation doesn't immediately appear separable because the terms involving both $x$ and $y$ are mixed in the first term.

Step 3: Check for an exact equation

An exact equation has the form $M(x, y) \, dx + N(x, y) \, dy = 0$, where $M(x, y) = 2x^2 + xy + y^2$ and $N(x, y) = 2x^2$, and the condition for exactness is:

$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$

Let's compute both partial derivatives:

• $\frac{\partial M}{\partial y} = \frac{\partial}{\partial y} (2x^2 + xy + y^2) = x + 2y$
• $\frac{\partial N}{\partial x} = \frac{\partial}{\partial x} (2x^2) = 4x$

Since $\frac{\partial M}{\partial y} = x + 2y$ and $\frac{\partial N}{\partial x} = 4x$, the equation is not exact.

Step 4: Attempt to find an integrating factor

If the equation is not exact, we can often make it exact by multiplying through by an appropriate integrating factor. In this case, finding an integrating factor might involve trial and error or specific techniques like recognizing a pattern in the variables.

Would you like to explore finding an integrating factor, or should we attempt a different method such as substitution?

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Follow-up Questions:

1. Do you want to explore how to find an integrating factor for this equation?
2. Would you like an explanation of exact equations in more detail?
3. Should we try using a substitution method, like $u = \frac{y}{x}$, to simplify the equation?
4. Are you familiar with separable differential equations, or would you like to see how we could check for this?
5. Should we focus on another type of method, such as finding a potential solution through approximation?

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Tip:

When dealing with non-exact equations, an integrating factor can often transform the equation into an exact one, making it solvable using standard techniques.