Solve for the eigenfunctions of the problem y''+2y'+(1-λ)=0, Y(0)=0, Y(1)=0?

Characteristic equation:
r^2 + 2r + (1 - λ) = 0
==> (r + 1)^2 = λ
==> r = -1 ± √λ.
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We have three cases:

(i) λ > 0.
Writing λ = k^2 for some k > 0, we have a general solution
y = e^(-x) (A cosh(kx) + B sinh(kx)).

y(0) = 0 ==> A = 0
y(1) = 0 ==> 0 = e^(-1)(0 + B sinh k) ==> B = 0.

So, y = 0 (ignore this case).
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(ii) λ = 0, which yields a double root of r = 0.
So, a general solution is y = A + Bx.

y(0) = 0 ==> A = 0
y(1) = 0 ==> 0 = 0 + B ==> B = 0.

So, y = 0 (ignore this case, too).
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(iii) λ > 0.
Writing λ = -k^2 for some k > 0, we have a general solution
y = e^(-x) (A cos(kx) + B sin(kx)).

y(0) = 0 ==> A = 0
y(1) = 0 ==> 0 = e^(-1)(0 + B sin k) ==> sin k = 0 ==> k = nπ for n = 1, 2, ... .

So, we have the eigenfunctions
y_n = B_n sin(nπx) for n = 1, 2, 3, ... [with λ = -(nπ)^2].
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I hope this helps!