The question is basically to find out the solution of the differential equation $$(2x^2+3y^2-7)xdx-(3x^2+2y^2-8)ydy=0$$.
Since the given differential equation is non homogeneous I tried to find out the point of intersection of $2x^2+3y^2-7=0$ and $3x^2+2y^2-8=0$.I got the point of intersection and then I tried to shift the coordinate system to that point.this removed the constant terms.However I was not able to proceed further.Please help me in this regard.thanks.
The solution (below) is obtained on the form of an implicit equation : $$(y^2-x^2+1)^5-c\:(y^2+x^2-3)=0$$ where $c$ is a constant.
Since it's a quintic polynomial equation, $y^2$ cannot be expressed as a function of $x^2$ with a finite number of elementary functions.
Theoretically, $y(x)$ could be expressed on a closed form thanks to special functions (in terms of Jacobi theta functions), but this would be very arduous : http://mathworld.wolfram.com/QuinticEquation.html
