Final answer:
By assuming a particular solution and applying boundary conditions to the general solution, we should arrive at a unique solution to the given boundary value problem.
Step-by-step explanation:
The boundary value problem you've presented consists of a second-order differential equation with specific boundary conditions. The given differential equation is y'' + 9y = 2e2x. The boundary conditions specify that y(0) = 0 and y(π) + y'(π) = 0. To find the solution, we'll assume a particular solution, yp(x), that has the form Ae2x for the non-homogeneous part and a complementary solution, yc(x), for the homogeneous equation y''+9y=0. The complementary solution typically has the form Bcos(3x) + Csin(3x). By plugging in the boundary conditions, we'll be able to solve for the constants A, B, and C, resulting in a unique expression that fulfills both the differential equation and the boundary conditions.
Considering the boundary conditions, the first one immediately sets B to 0 since cos(0) = 1 and we require y(0) = 0. The second boundary condition will provide an equation involving both A, and C, which together with the differential equation will allow us to solve for these constants. It is this systematic approach of applying the boundary conditions to the general solution that enables us to solve boundary value problems.
In conclusion, we should be able to find a unique solution to the boundary value problem by following this method and satisfying the given conditions with specific values for A, B, and C.