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))`
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