How Do You Solve the Differential Equation kxy = (y+dx)(x+dy)?

[r16]

I have run into this problem solving differential equations of this type (they occur often doing momentum problems):

$$kxy = (y+dx)(x+dy)$$

where $k$ is constant. I multiply it out to :

$$kxy= xy + xdx + ydy + dydx$$

Regroup and :

[tex]\int {kxy} = \int {xdx} + \int {ydy} + \int {dydx} [/itex] <br /> <br /> I'm left with the term $\int dxdy$ that I don't know what to do with. Am I able to hold either the $dx$ or $dy$ constant and integrate with respect to the other? I am not able to find a transformation that will remove the $dydx$ or $\frac{dy}{dx}$ or $\frac{dx}{dy}$. I am also confused about the term $\int kxy$: integration without respect to a particular differential. How would I solve this differential equation?[/tex]

 

[HallsofIvy]

There something wrong with your equation. You can't have "dx" and "dy" mixed like that. If you are dealing with "differentials" dx and dy, it might make sense (but it would just say kxy= xy) but it is certainly not a differential equation.