What is the solution of dy/dx = (x^2+y^2)/(xy)

The differential equation `dy/dx = (x^2+y^2)/(xy)` has to be solved.

`dy/dx = (x^2+y^2)/(xy)`

This can be written as

`dy/dx = x^2/(xy) + y^2/(xy)`

`dy/dx = x/y + y/x`

`dy/dx = (y/x)^-1 + y/x`

Let `f = y/x`

`y = f*x`

`dy/dx = f + x*(df)/(dx) `

Now substituting for `dy/dx` and `y/x` in the equation we get

`f + x*(df)/(dx) = f^-1 + f`

`x*(df)/(dx) = 1/f`

`f*df = dx/x`

Take the integral of both the sides

`int f*df = int dx/x`

`f^2/2 = ln x + C`

The constant C can be included in the logarithm of x as `ln(k*x)`

`(y/x)^2/2 = ln(k*x)`

`y^2 = 2*x^2*ln(k*x)`

`y = +-sqrt(2*x^2*ln(k*x))`

The solution of the differential equation `dy/dx = (x^2+y^2)/(xy)` is `y = +-sqrt(2*x^2*ln(k*x))`