dy dx

The given differential equation is:

$\frac{dy}{dx} = \frac{y^3 + y}{x^3 + x}$

Let's analyze this equation step by step based on the types of equations you mentioned:

1. Separable:

A differential equation is separable if it can be written as the product of functions of $y$ and $x$. Specifically, it would look like:

$\frac{dy}{dx} = f(y) \cdot g(x)$

In this case, we cannot separate the equation into such a product because the equation involves higher powers of $y$ (i.e., $y^3 + y$) and $x$ (i.e., $x^3 + x$), and we cannot isolate a function of $y$ on one side and a function of $x$ on the other.

So, the equation is not separable.

2. Exact:

A differential equation is exact if it can be written in the form:

$M(x, y) dx + N(x, y) dy = 0$

where $M(x, y)$ and $N(x, y)$ satisfy the condition:

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

This equation doesn't quite fit this form, and there is no immediate way to express it in exact form, so the equation is not exact.

3. Linear in $x$:

A differential equation is linear in $x$ if the dependent variable $y$ and its derivatives appear to the first power, and no products of $x$ and $y$ or higher powers of $y$ are involved.

This equation involves $y^3$, which is a cubic term. So it is not linear in $x$.

4. Linear in $y$:

A differential equation is linear in $y$ if $y$ appears to the first power and there are no terms involving powers of $y$ greater than 1, such as $y^2$ or $y^3$.

Since this equation involves $y^3$, it is not linear in $y$.

5. Homogeneous:

A differential equation is homogeneous if the degree of the terms in $x$ and $y$ are the same (or, in simpler terms, if the equation can be written in such a way that every term is of the same degree in $x$ and $y$).

Here, the degree of $y^3 + y$ is 3 (since $y^3$ is the highest degree term), and the degree of $x^3 + x$ is also 3. Since the degrees match, the equation is homogeneous.

6. Bernoulli in $x$ and Bernoulli in $y$:

A Bernoulli equation is of the form:

$\frac{dy}{dx} + P(x)y = Q(x)y^n$

where $n \neq 1$. The given equation does not fit this form because it does not involve terms that can be simplified to such a structure, and we have powers of $y$ that are more complicated than just $y^n$ (it has a cubic term). So it is not Bernoulli in $x$ or Bernoulli in $y$.

Conclusion:

The given equation is homogeneous.

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Would you like a more detailed explanation of any of these categories or the steps involved in analyzing the equation? Here are some related questions you might find useful:

1. What does it mean for a differential equation to be linear in $x$ or $y$?
2. How can you transform a homogeneous equation into a simpler form?
3. What methods can be used to solve a homogeneous differential equation?
4. How does one determine if a differential equation is exact?
5. What is the process for solving Bernoulli's equation?

Tip: When determining if a differential equation is homogeneous, check if the degrees of the highest powers of $x$ and $y$ match.