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

tickets for the harlem globetrotters

show at michigan state university in

2010 cost $16, $23, or, for vip seats, $40.

total revenue = 16x + 23y + 40z

if 9 times as many $16 tickets were sold

as vip tickets,

x = 9z => x – 9z = 0, if 9 times as many $16 tickets were sold

as vip tickets,

x = 9z => x – 9z = 0, sales of all three kinds of tickets would

total $46,575.

16x + 23y + 40z = 46575, how many of each kind of ticket would

have been sold?

solve for x, y and z

x − 9z = 0

x − y − z = 55

16x + 23y + 40z = 46575

x = 1170

y = 985

z = 130

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