Question

If 2xy3 +3ycos(xy)+(cx2y2 +3xcos(xy))y=0 is an exact equation, what isthe value of c?

Answer 1

$\text{The given equation can also be written as } \\(2xy^3 + 3y\cos xy)dx + (cx^2y^2+3x \cos xy)dy \\\text{Let $u=2xy^3 + 3y\cos xy$ and $v=cx^2y^2+3x \cos xy$} \\\text{Next, we differentiate u with respect to y and v with respect to y} \\\text{Therefore, } \frac{\partial u}{\partial y} = 6xy^2+ 3\cos xy - 3xy \sin xy \\\frac{\partial v}{\partial x} = 2cxy^2+ 3\cos xy - 3xy \sin xy \\\text{For the differential equation to be exact $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial x}$} \\\text{Hence comparing both equations we have that} \\2c =6 \implies c = 3$