Consider the wave equation Utt- c²Uxx = 0 ᏆᏆ the half-line x = [0, ∞) with boundary condition U (0, t) = 0 d initial conditions U(x,0) = f(x), Ut(x,0) = g(x). this problem we construct the solution to the above problem using even extensions. For s, let F(x) = { √(-²2) +²0 + f(x) x>0 1

If r(t) = cos(5t)i + sin(5t)j + Otk compute r'(t)= i+ j+ k and ſr(t) dt= i+ j+ k + C with C a constant vector.

Use Green's Theorem to evaluate Se xy dx + (1 – æy³) dy where C is shown below y (1,4) (-1,2) 2 1 (1,1) (-1,-1) -1

The two-dimensional Laplace equation, f(x,y) = 5x-9y-6 2²4 2²4 -= 0, describes potentials and steady-state temperature distributions in a plane. Show that the following function satisfies the two-dimensional Laplace equation. dx