Can an differential equation be a member of many classes of differential equations? I just started learning ODE online and I'm not sure I fully understand when I should use one method or the other.

For example, I know that $\frac{dy}{dx}=\frac{y}{x}$is separable, but is it homogeneous? I tried to solve it using the homogeneous approach but I get nonsense. Another example is bernoulli equations, like $\frac{dy}{dx}-\frac{y}{x}=xy^{n}$  whenever $n=1$ I also get nonsense, can anyone help me?

 A: Of course an Ordinary Differential Equations (ODE) can be solved using different approach. On the other hand if you are studying a first order differential equations so you will learn:

*

*Separable Differential Equations:
$$\color{blue}{\boxed{\frac{dy}{dx}=f(x)g(y)}}.$$

*Linear Differential Equations:
$$\color{blue}{\boxed{\frac{dy}{dx}+P(x)y=f(x)}}.$$

*Exact Differential Equations:
$$\color{blue}{\boxed{M(x,y)dx+N(x,y)dy=0 \quad \text{with} \quad \frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}}}.$$

*Bernoulli's Differential Equations
$$\color{blue}{\boxed{\frac{dy}{dx}+P(x)y=f(x)y^{n}, \quad n\in \mathbf{R}}}.$$
Here if $n\not=0, 1$ the substitution $u=y^{1-n}$ reduce the Bernoulli's Differential Equation to Linear Differential Equation.

*Homogeneous Differential Equation:
$$\color{blue}{\boxed{\frac{dy}{dx}=f\left(\frac{a_{0}x+a_{1}y+a_{3}}{b_{0}x+b_{1}y+b_{3}}\right)}}$$
If you are solving the Ordinary Differential Equation
$$\frac{dy}{dx}=\frac{y}{x}$$
so you can solve the ODE using  $1)$, $2)$ and $5)$.
Indeed,

*

*$\displaystyle \frac{dy}{dx}=\frac{y}{x} \implies \int \frac{1}{y}dy=\int \frac{1}{x}dx \implies \ln|y|=\ln|x|+c \implies y=cx, \quad x>0.$

*$\displaystyle \frac{dy}{dx}=\frac{y}{x} \implies \frac{dy}{dx}-\frac{1}{x}y=0 \implies  x\left( \frac{dy}{dx}-\frac{1}{x}y\right)=0, \quad x>0 \implies y=cx, \quad x>0.$

*$\displaystyle \frac{dy}{dx}=\frac{y}{x} \overset{u=y/x}{\implies}u+x\frac{du}{dx}=u \implies y=cx, \quad x>0.$
