How do i solve this ODE?

1. Sep 17, 2014

gahara31

1. The problem statement, all variables and given/known data
the first one
y'=$\frac{y^{2}+xy^{2}}{x^{2}y-x^{2}}$

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

3. 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

2. Sep 17, 2014

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.

3. Sep 18, 2014

gahara31

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

4. Sep 19, 2014

HallsofIvy

Staff Emeritus
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!