Find a second independent solution y1=x xy"-xy'+y=0

We can use reduction of order. Given that is a known solution, we look for a solution of the form . It has derivatives and . Substituting these into the ODE gives

Let so that and we get an ODE linear in :

Divide both sides by :

, we can condense the left side as the derivative of a product:

Integrate both sides and solve for :

Integrate both sides again to solve for . Unfortunately, there is no closed form for the integral of the right side, but we can leave the result in the form of a definite integral:

where is any point on an interval over which a solution to the ODE exists.

Finally, multiply by to solve for :

already accounts for the term above, so the second independent solution is

water molecules will move from the side of higher water concentration to the side of lower concentration until both solutions are isotonic. at this point, equilibrium is reached. in an isotonic solution, the flow of water in and out of the cell is happening at the same rate.

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