Show that the boundary value problem d dx (p(x)y ') q(x)y f (x), 0 x 1, y(0) , y(1) , can be transformed by the change of variable z y x (1 x) into the form d dx (p(x)z ') q(x)z F(x), 0 x 1, z(0) 0, z(1) 0 ...

With zx yx x 1 x we have z0 y0 a a a 0 and z1 y1 b b b 0 ...

Show that the boundary-value problem d / dx (p(x)y') + q(x)y = f (x), 0 x

Question:

Show that the boundary-value problem
− d / dx (p(x)y') + q(x)y = f (x), 0≤ x ≤ 1, y(0) = α, y(1) = β,
can be transformed by the change of variable
z = y − βx − (1 − x)α
into the form
− d / dx (p(x)z') + q(x)z = F(x), 0≤ x ≤ 1, z(0) = 0, z(1) = 0.