<p>EGR 599 ______LAST NAME, FIRST Problem set #8</p><p>1. Use the solution u(x,t) = F(x + ct) + G(x  ct) to find the function that satisfies the wave u equation and the initial conditions u(x,0) = 0, (x,0) =  2xexp(x2). t</p><p>2. Use the solution u(x,t) = F(x + ct) + G(x  ct) to find the function that satisfies the wave 1 u equation and the initial conditions u(x,0) = , (x,0) =  2xexp(x2). 1  x 2 t</p><p>3. Let f(x) denote the shape of a plucked string of length p with endpoints fastened at x = 0 and x = p, as shown. y</p><p> h</p><p> x 0 a p a) Obtain the sine series expansion of f(x). b) Let a = 1/3, p = 1, and h = 1/10 and plotting the resulting function f(x) with 2 and 20 terms partial sums. </p><p>4. Determine whether the given partial differential equation and boundary conditions are linear or nonlinear, homogeneous or nonhomogeneous, and the order of the PDE. a) uxx + uxy = 2u, ux(0, y) = 0. b) uxx  ut = f(x, t), ut(x, 0) = 2. c) utux + uxt = 2u, u(0, t) + ux(0, t) = 0.</p><p>1 5. Verify that the given function u = x 2  y 2  z 2</p><p> is a solution of the three dimensional Laplace equation uxx + uyy + uzz = 0</p><p> 2u  2u 6. Solve = c2 c = 1, L= 1 t 2 x 2</p><p>The boundary and initial conditions required for the solution of the wave equations are</p><p>B.C. : u(0,t) = 0 and u(L,t) = 0, for t  0 I.C. : u(x,0) = f(x) = sin x + 3sin 2x  sin 5x and u (x,0) = g(x) = 0, for 0  x  L t</p><p> 2u  2u 7. Solve = c2 c = 4, L= 1 t 2 x 2</p><p>The boundary and initial conditions required for the solution of the wave equations are</p><p>B.C. : u(0,t) = 0 and u(L,t) = 0, for t  0</p><p> 2x if 0  x  1/ 2 I.C. : u(x,0) = f(x) =  and 2(1  x) if 1/ 2  x  1 u (x,0) = g(x) = 0, for 0  x  L t</p><p>8. In what regions are the following PDEs elliptic, hyperbolic or parabolic?</p><p> 2u  2u (a) + 4 = 0, x 2 y 2  2u u (b) 7  3 = 0, xy y  2u  2u  2u u (c) x2 + 4y + + 2 = 0, x 2 xy y 2 x  2u  2u (d) 3y  x = 0, x 2 y 2  2u u (e) + 4 = 0, x 2 x  2u  2u  2u (f) x2y2 + 2xy + = 0. x 2 xy y 2</p><p>9. Use d’Alembert’s formula </p><p>1 1 xct u(x,t) = [f*(x  ct) + f*(x + ct)] + g * (s)ds 2 2c xct</p><p> 2u  2u to solve = c2 c = 1, L= 1 t 2 x 2</p><p>The boundary and initial conditions required for the solution of the wave equations are B.C. : u(0,t) = 0 and u(L,t) = 0, for t  0 u I.C. : u(x,0) = sin x + 3 sin 2x and (x,0) = sin x, for 0  x  L t</p>