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**Solve y' = y/x + 1/y**

I get a similar answer to the correct one but I believe I am making a substitution error. Here is my attempt:

dy/dx = y/x + 1/y

set v = y/x

equation now becomes: v + x(dv/dx) = v + 1/(x*v)

reduces to: dv/dx = 1/(x^2 * v)

Now the equation is seperable, so I seperate and take the integral of both sides yielding:

v = (sqrt(-2) * sqrt(x*c - 1))/sqrt(x)

--even if i substitute v = y/x back in it doesn't come out to be the correct solution of:

y = sqrt(x) * sqrt(x*c -2)

dy/dx = y/x + 1/y

set v = y/x

equation now becomes: v + x(dv/dx) = v + 1/(x*v)

reduces to: dv/dx = 1/(x^2 * v)

Now the equation is seperable, so I seperate and take the integral of both sides yielding:

v = (sqrt(-2) * sqrt(x*c - 1))/sqrt(x)

--even if i substitute v = y/x back in it doesn't come out to be the correct solution of:

y = sqrt(x) * sqrt(x*c -2)

Any insight would be great!