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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^2-2y 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.