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:

2

y 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:

2

y dy+d(xy) = 0

Integrating gives us:

y^{2}+xy=K(General Solution)

**CHECK**

We obtained the solution *y*^{2} + *xy *=* K*. Is it correct?

Now

`d/dx y^2=2y dy/dx` (using Implicit Differentiation),

`d/dx xy=x dy/dx + y` (using product rule).

and `d/dx(k)=0`.

So putting this together gives the derivative of our answer as:

`2y(dy)/(dx)+x(dy)/(dx)+y=0`

On multiplying by *dx* throughout, we have:

2

y dy+x dy+y dx= 0

which is the DE that we started with so we know our general solution is correct.