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# Find the solution of the differential equation xy′′ – y′ + (1 – x)y = 0

Assume the differential equation

has a solution of the form

for some constant . Determine an explicit formula for this solution.

Incomplete.

1. Tyler says:

Evangelos has a great solution. But I think I found a faster solution (If somewhat possibly incomplete)

If you plug in x=0 (x=1 works similarly) we get y’=y, so e^x immediately follows as a solution.

2. Evangelos says:

Since we are given

We then know that

And we have

Since e^mx can never be equal to zero, we can then divide both sides by it

But that would mean that
m^{2} – 1 & = 0 \\
1 – m &= 0
\end{align*}

Thus, m = 1, and we have

As requested.