Final answer:
To solve the given boundary-value problem, we need to find the function y(x) that satisfies the given differential equation y + 81y = 0, and the boundary conditions y(0) = 0 and y(π) = 0. The solution to the boundary-value problem is y(x) = 0.
Step-by-step explanation:
To solve the given boundary-value problem, we need to find the function y(x) that satisfies the given differential equation y + 81y = 0, and the boundary conditions y(0) = 0 and y(π) = 0.
This is a linear homogeneous differential equation with constant coefficients, so we can solve it by assuming the solution takes the form y(x) = e^(rx) and substituting it into the equation. We obtain r^2 + 81 = 0, which gives us r = ±9i.
Since the boundary conditions are given, we can write the general solution as y(x) = c1*cos(9x) + c2*sin(9x), where c1 and c2 are constants. Using the given boundary conditions, we find c1 = 0 and c2 = 0. Therefore, the solution to the boundary-value problem is y(x) = 0.