Determine the specific solution for the nonhomogenous PDE Th

Determine the specific solution for the non-homogenous PDE. Thank you!

ut uara (t t 2) e t cos(r) (t3 2) cos(2a), ua (0, t a (T, t) 0 u T, 0) 10 cos(z) 2 cos (2a) cos(6r), 0 S t 00

Solution

We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard form y + p(t) y + q(t) y = g(t), g(t) 0. (*) Each such nonhomogeneous equation has a corresponding homogeneous equation: y + p(t) y + q(t) y = 0. (**) Note that the two equations have the same left-hand side, (**) is just the homogeneous version of (*), with g(t) = 0. We will focus our attention to the simpler topic of nonhomogeneous second order linear equations with constant coefficients: a y + b y + c y = g(t). Where a, b, and c are constants, a 0; and g(t) 0. It has a corresponding homogeneous equation a y + b y + c y = 0.