DSolve Ode's with boundary condition

Question

Equations

T''[r] + T'[r] + a/k1==0

U''[r] + U'[r]==0

Boundary Conditions

T'[0]==0

T[r1]==U[r1]==Tw

k1*T'[r1]==k2*U[r2]

U[r2]==Ts

What would be the simplest way of solving for T and U in mathematica? I can do it in individual steps as I would do on paper, but I feel like there should be an easier way.

Answer 1

Because there is one too many boundary conditions, I chose to omit k1*T'[r1]==k2*U[r2] for now. Then,

Flatten@DSolve[{T''[r] + T'[r] + a/k1 == 0, U''[r] + U'[r] == 0, T'[0] == 0, 
    U[r2] == Ts, T[r1] == U[r1] == Tw}, {U, T}, r]

(* {T -> Function[{r}, (E^(-r - r1) (a E^r - a E^r1 - a E^(r + r1) r + a E^(r + r1) r1 + 
         E^(r + r1) k1 Tw))/k1], 
    U -> Function[{r}, -((E^-r (-E^(r + r2) Ts + E^(r1 + r2) Ts + E^(r + r1) Tw - 
         E^(r1 + r2) Tw))/(-E^r1 + E^r2))]} *)

Applying this to the final boundary condition yields

Simplify[(k1*T'[r1] == k2*U[r2]) /. %]

(* a + k2 Ts == a E^-r1 *)

which is a constraint on the constants.