Is this DE separable or homogeneous?

Question

Differential Eq is $$\frac{(3y^2-t^2)}{y^5} \frac{dy}{dt} + \frac{t}{2y^4}=0$$

Is it linear in $y$ or $t$? Is it Bernoulli in $y$ or $t$?

Answer 1

$$\frac{dt}{dy}=6\frac{y}{t}-2\frac{t}{y}\qquad (1)$$ This is an homogeneous ODE which can be transformed to a linear ODE with the change of variable : $y=t\:x$. The transformed ODE is linear for the function $t(x)$.

The ODE $(1)$ is also of the Bernoulli kind : $$\frac{dt}{dy}+\frac{2}{y}t=6y\:t^n\quad \text{with }n=-1$$ where the unknown is $t(y)$. The change of function $f(y)=\left( t(y)\right)^2$ transforms it to a linear ODE.

So, the ODE $(1)$ is both of the homogeneous and Bernoulli kinds. So you have two methods to solve it.