Does this differential equation have an elementary solution? $$\frac{dx}{dy} =x-y^2$$
My professor told the class that this differential equation has no solutions. My question is how come? can't you just separate variables shown here. 
$$y^2dy =xdx$$
$$y= \sqrt[3]{ \frac{x^2}{6} } $$
 A: Your answer is wrong but $x(y)=e^{y} (c-\int_0^{y} t^{2}e^{-t} dt)$ is a solution of the given DE for any constant $c$. Using integration by parts we can write the solution as $x=ce^{y}+y^{2}+2y+2$.
Method used to solve the equation: multiply by the interagting factor $e^{-y}$ to write the equation as $(e^{-y}x)'=-y^{2} e^{-y}$ and integrate. 
A: This is an easy linear equation.
The homogeneous solution is obviously $x=Ce^y$, while a particular solution can be a quadratic polynomial.
$$x=ay^2+by+c\to2ay+b=ay^2+by+c-y^2$$ yields $$a=1,b=c=2$$ and $$x=y^2+2y+2.$$
Hence the general solution
$$\color{green}{x=y^2+2y+2+Ce^y}.$$
A: $$\frac{dx}{dy} =x-y^2$$
The solution of this ODE is :
$$x(y)=ce^y+y^2+2y+2$$
Probably your Professor didn't say "this differential equation has no solution" but he said something that you not well understood.
Possibly he was talking of the function $y(x)$ which is the inverse function of the above known function $x(y)$. 
In fact the function $y(x)$ exists but cannot be written in terms of a finite number of elementary functions. So one cannot write it explicitly even though $x(y)$ is already found explicitly.
Another eventality might be a typo in the ODE. Can't the ODE be :
$$\frac{dy}{dx} =x-y^2$$
This is an ODE of Riccati kind which is solvable in terms of Airy special functions. In this case the Professor would say that the solutions exist but cannot be expressed with a finit number of the usual elementary functions.
