## Differential Equations

I have the following questions. My due date is on 3rd march.

1. What function u(x) with u(0) = 0 and u(1) = 0 minimizes P(u) = ∫[½ (du/dx)2 + x u(x)] dx ?
(The integral is from 0 to 1)

2. What function w(x) with dw/dx = x (with unknown integration constant) minimizes
Q(w) = ∫(w2/2) dx ? (The integral is from 0 to 1)

3. What functions u and w minimize P and Q with dw/dx = x and u(0) = w(1) = 0? Verify the
strong duality –P = Q.

4. From the differential equation –d/dx(c du/dx) = f, derive the weak form (4) by multiplying by
test functions v and integrating.

5. Show that P(u) + Q(w) > 0 for any admissible uw: and
∫[(c/2) (du/dx)2 – fu + (1/(2c)) w2] dx > 0 when f = – dw/dx and u(0) = w(1) = 0.
(Integral is from 0 to 1)

6. Show that P(u) + Q(w) = 0 for the optimal u and w, which satisfy w = cu

Any help is welcome
Thanks