Solve differential equation \frac{dy}{dx}= \frac{x}{y}, \ y(0)= -8

My question is to find the solutions to the following
$\frac{df(x)}{dx}=f^{-1}(x)$
where $f^{-1}(x)$ refers to the inverse of the function f. The domain really isn't important, though I am interested in either (-inf, inf) or (0, inf), so if any solutions are known for more restricted domains then they are welcome.
I cannot find any material relating to this type of question in any of my calculus and differential equations textbooks and references; it seems quite unorthodox. Any material which covers this type of diff equation would be wlecome

Solve differential equation ${\displaystyle y^{\prime }-3x^{2}y=e^{x^3}}$

I was trying to compute the solution for the following differential equation:
$x(2x^{2}ylog(y)+1)y^{\prime }=2y$
As I couldn't get anywhere I checked the hints in the textbook which are the following:
Reverse the way of thinking, namely view $x$ as a function and $y$ as a variable, considering that
$y=\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}=>y^{\prime }=\frac{1}{x^{\prime }}$. Then it goes to say that the equation now becomes
$x^{\prime }-\frac{x}{2y}=log(y)x^3$
This final equation is obviously simple enough to solve, but how on Earth did they arrive there?

Let ${\displaystyle (\bar{x},\bar{y},\bar{z})}$ be an equilibrium of autonomous first-order differential equation:
${\displaystyle \{\begin{array}{c}\dot{x}=f_1(x,y,z)\\ \dot{y}=f_2(x,y,z)\\ \dot{z}=f_3(x,y,z)\end{array}}$
Is it possible to say that ${\displaystyle (\bar{x},\bar{y},\bar{z})}$ is unique equilibrium point of the following system:
${\displaystyle \{\begin{array}{c}\dot{x}=f_1(x,y,z)-(x-\bar{x})\\ \dot{y}=f_2(x,y,z)-(y-\bar{y})\\ \dot{z}=f_3(x,y,z)-(z-\bar{z})\end{array}}$

I really dont know what to do whit this equation, i have tried to integrate it, but i can do it.
${\displaystyle y^{\prime 3}-y{y}^{\prime 2}-x^2{y}^{\prime }+x^{2}y=0}$

Suppose that a dose of ${\displaystyle x_0}$ (initial) gram of a drug is injected into the bloodstream. Assume that the drug leaves the blood and enters the urine at a rate proportional to the amount of drug present in the blood. In addition, assume that half of the drug dose has entered the urine after 0.75 hour. Find the time at which the amount of drug in the blood stream is 5% of the original drug dose ${\displaystyle x_0}$ (initial).
I have the first order differential equation. It's finding ${\displaystyle t}$ that is the problem.

Solve differential equation $t{y}^{\prime }+(t+1)y=t,y(\ln 2)=1$, t>0