Suppose the solution of the boundary-value problem

y'' + Py' + Qy = f(x), y(a) = 0, y(b) = 0,

a < b, is given by y_p(x) = ∫^b_aG(x,t)f(t)dt where y₁(x) and y₂(x) are solutions of the associated homogeneous differential equation chosen in the construction of G(x, t) so that y₁(a) = 1 and y₂(b) = 0. Prove that the solution of the boundary-value problem with nonhomogeneous DE and boundary conditions,

y'' + Py' + Qy = f(x), y(a) = A, y(b) = B

is given by

In your proof, you will have to show that y₁(b) ≠ and y₂(a) ≠ 0. Reread the assumption following (24).

Transcribed Image Text:

B y(x) = y,(x) + A + (x)' la2(r).