Final answer:
To find all λ > 0 such that the boundary value problem has a solution other than y(x) = 0, we analyze the given second-order linear homogeneous differential equation with constant coefficients. The solution will be in the form y(x) = c1e^(√λx) + c2e^(-√λx), where c1 and c2 are constants.
Step-by-step explanation:
To find all λ > 0 such that the boundary value problem has a solution other than y(x) = 0, we start by considering the given problem:
y′′ + λy = 0, y(0) = y(L) = 0
This is a second-order linear homogeneous differential equation with constant coefficients. The characteristic equation is obtained by assuming the solution of the form y(x) = e^(rx), where r is a constant:
r^2 + λ = 0
Solving this equation, we find the roots:
r = ±√(−λ)
Since we want a non-trivial solution (other than y(x) = 0), we want the roots to be non-zero. Hence, we need −λ < 0, or in other words, λ > 0.
Therefore, all λ values greater than 0 will have solutions other than y(x) = 0. The solution in this case will be of the form:
y(x) = c1e^(√λx) + c2e^(−√λx), where c1 and c2 are constants.