Solve the boundary value problem y4y5y0 y00y10 Determine if

Solve the boundary value problem y\'\'+4y\'-5y=0, y(0)=0,y(1)=0. Determine if the solution is unique, represents an infinite set of solution,or does not exist.

Solution

m^2 +4m -5 = 0 m = -5 or 1 y = c1e^(-5x) + c2e^x y(0) = c1 + c2 = 0 y\' = -5c1e^(-5x) + c2e^x y\'(0) = -5c1 + c2 = 1 6c1 = -1, c1 = -1/6, c2 = 1/6 y = -(1/6)e^(-5x) + (1/6)(e^x) unique solution