Solving 2nd Order Differential Equation with Dirichlet BCs

[dirk_mec1]

Homework Statement

Solve:

$$-D(x) \frac{d^2 T}{dx^2}=1$$

for $$x \in [0,1]$$

D(x) =10^-3 in [0,0.5] and D(x) = 1 in (0.5,1]

with homogeneous dirichlet boundary conditions

The Attempt at a Solution

So I have two quadratic equations with x(0)=x(1)=0 and continuity at x=0.5 but I'm missing a BC. I thought of the derative but I am uncertain. Can someone help me?

 

[HallsofIvy]

Missing a Boundary Condition? There are no boundary conditions given in the problem at all. You are asked for the general solution.

Solve for the general solution. T₁(x), of $d^2T/dx^2= -1000$ between 0 and 0.5. That answer will involve two unknown constants, say C and D. Then solve for the general solution, T₂(x), of [itex]d^2T/dx^2= -1 between 0.5 and 1. That will involve two new constants, say E and F. Set the values of the functions and their first derivatives equal at 0.5 in order to write E and F in terms of A and B. You should then have a two "piece" definition for T(x) both involving the same two constants, A and B.