You are close!
The form of the differential equation suggests a check for exactness.
(Df/Dx)dx + (Df/Dy)dy = 0 where 'D' is used as the symbol for partial differentiation.
Check if D(Df/Dx)/Dy = D(Df/Dy)/Dx. In this case 2x + 3y^2 = 2x + 3y^2, so the differential equation is exact.
Df/Dx = 2xy + y^3 is to be integrated with respect to x to get f(x,y) = yx^2 + xy^3 + f(y)
Df/Dy = x^2 + 3xy^22y integrates to f(x,y) = yx^2 +xy^3  y^2 + f(x)
Comparing the two solutions, f(y) = y^2 and f(x) = 0.
The solution is C = yx^2 +xy^3 y^2. Although we work with partial derivatives, exact differential equations are usually taught in the differential equations.
In some college math departments, linear algebra is taught along with differential equations in the same class. Some colleges require linear algebra or differential equations or both depending on a student's major. The differential equations class usually follow a 2 or 3 semester sequence in calculus. A basic linear algebra course could be taught that does not require prerequisites in calculus. The differential equations class usually covers DE's with one independent variable (ordinary differential equations). Maybe at the end of the course, some partial differential equations are introduced but are usually covered in an advanced courses.
6/17/2013

Francisco P.
Comments
Do you guys know how to input this equation on Ti89 calculator to find the answer? I know how to do this problem but I want to know this technique to check other problems. Like you press F3, then C, deSolve...How do I put this equation?
I don't have T89 in my hands now, but Ias i rememeber, you have to go to MATH menu. It opens a llist of options. Scroll down the screen to get to to the line "differential equations". As I know, it opens the screen in which you can type your equation.