First, we note that we cannot separate the variables such that the x expressions would be with dx only and the y expressions would be with dy only. So the DE is not separable.
So we aim to get the DE in one of the integrable combination forms. Rearranging, we have:
2y dy + (x dy + y dx) = 0
From the integrable combination (1) given above, the expression in brackets can be written as d(xy) and the differential equation is equivalent to:
2y dy + d(xy) = 0
Integrating gives us:
y2 + xy = K (General Solution)
We obtained the solution y2 + xy = K. Is it correct?
`d/dx y^2=2y dy/dx` (using Implicit Differentiation),
`d/dx xy=x dy/dx + y` (using product rule).
So putting this together gives the derivative of our answer as:
On multiplying by dx throughout, we have:
2y dy + x dy + y dx = 0
which is the DE that we started with so we know our general solution is correct.