Question

Find all the real eigenvalues and eigenfunctions to the following problem:

y''+4 y'+λ y=0, y(0)=0, y(π)=0

Answer 1

To find the real eigenvalues and eigenfunctions for the given problem y'' + 4y' + λy = 0 with boundary conditions y(0) = 0 and y(π) = 0, we can solve the characteristic equation and substitute the values of r into the original assumed solution.

Step-by-step explanation:

To find the real eigenvalues and eigenfunctions for the given problem, we can solve the differential equation y'' + 4y' + λy = 0 with the boundary conditions y(0) = 0 and y(π) = 0.

First, we assume a solution of the form y(x) = e^(rx), where r is a constant. Plugging this into the differential equation gives us the characteristic equation r^2 + 4r + λ = 0.

Solving the characteristic equation, we find two values of r: r₁ = (-4 + √(16 - 4λ))/2 and r₂ = (-4 - √(16 - 4λ))/2. These correspond to the eigenvalues of the problem.

To find the eigenfunctions, we substitute the values of r into the original assumed solution and solve for y(x). The eigenfunctions are given by y(x) = A₁e^(r₁x) + A₂e^(r₂x), where A₁ and A₂ are constants determined by the boundary conditions.