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The problem says:

Given that [tex]c1e^x+c2e^{-x}[/tex] is a 2 parameter family of solutions of

[tex]xy''-y' = 0[/tex] on the interval(-infinity, infinity), show that constants c1 and c2 cannot be found so that a member of the family satisfies the initial conditions y(0) = 0, y'(0) = 1. Explain why this does not violate theorem 4.1

I guess part of my problem is that I'm not really clear what the difference is between a family of solutions and a unique solution. Is it just the difference between y=x+C and y=x+4 where C=4?

But back to the problem at hand, given those initial conditions, it looks like

c1 = 0 and c2 can't be found. Would that be the majority of my answer, take the derivative of the solution and show that c2 can't be found?

I'm also not sure what they mean, "does not violate theorem 4.1".

Theorem 4.1 says

[tex] let a_{n}(x), a_{n-1}(x),.......a_1(x),a_0(x) and g(x)[/tex] be continuous on an interval I and let [tex]a_n(x) not equal 0[/tex] for every x in this interval. If [tex]x = x_0[/tex] is any point in this interval, then a solution y(x) of the IVP exists on the interval and is unique.