How can I simplify these ODEs?

[gahara31]

Homework Statement

the first one
y'=$\frac{y^{2}+xy^{2}}{x^{2}y-x^{2}}$

the second one
xyy'=$\frac{x^{2}+1}{y+1}$

Homework Equations

The Attempt at a Solution

i separated x and y variable then integrate both of them

in the first one
∫$\frac{y-1}{y^{2}}$dy=∫$\frac{1+x}{x^{2}}$dx

ln|y|+$\frac{1}{y}$+C=- $\frac{1}{x}$+ln|x|+C

and the second one
∫y(y+1)dy = ∫$\frac{x^{2}+1}{x}$dx

$\frac{y^{3}}{3}$+$\frac{y^{2}}{2}$+C=$\frac{x^{2}}{2}$+ln|x|+C

but i can't change both of them into f(x) form or any simpler form

 

[slider142]

It is rare that you will find a differential equation with a solution that can be written as an explicit function. Implicit solutions, the equations relating x and y that you found, are usually accepted as finding a solution to a differential equation as well. As long as there are no derivatives in your final equation, and you specify the domain of the implicit function y that is defined by your equation, where it satisfies the original differential equation, you have found a solution.
Note, however, that you do not need two constants of integration: you may condense them into a single constant: C₁ - C₂ = C.

 

[gahara31]

It is rare that you will find a differential equation with a solution that can be written as an explicit function. Implicit solutions, the equations relating x and y that you found, are usually accepted as finding a solution to a differential equation as well. As long as there are no derivatives in your final equation, and you specify the domain of the implicit function y that is defined by your equation, where it satisfies the original differential equation, you have found a solution.
Note, however, that you do not need two constants of integration: you may condense them into a single constant: C₁ - C₂ = C.

i see, i just don't really understand the difference between implicit and explicit form, so the thing i just solve is the implicit form.. thanks for answering

 

[HallsofIvy]

The only thing to "understand" about "implicit" and "explicit" form is that the explicit form is always "y= some expression in x only" and the implicit form isn't!