PDE's

Please help me skech the graphs. I know how to sketch a graph of cosx, but other then that, I'm confused and I don't have a math program to just plug in the numbers to see what it looks like...Please see attached. Thank you!

Wave Equation

For a solution of the wave equation with p=T=C=1 the energy density is defined as e=1/2 (U_t ^2 + U_x ^2) and the momentum density as p=U_t*U_x Show that de/dt=dp/dx and dp/dt=de/dx Show that both e(x,t) and p(x,t) also satisfy the wave equation http://tosio.math.toronto.edu/pdewiki/index.php/2006APM346Midterm1 It's problem number 4 from this site.

PDE's - trigonometric system

I would like to see how these problems are solved. Thank you.

Diffusion Equation

Show that S(x,y,t)=S(x,t)S(y,t) satisfies the diffusion equation. S_t = k(S_xx + S_yy)

PDE's - Solve the wave equation

I would like to see how these problems are solved. Thanks. 3. Solve the wave equation, ∂2u/∂t2 = c2(∂2u/∂x) -∞ < x < ∞ With initial conditions, u(x,0) = (1/x2+1)sin(x), and ∂u/∂t(x,0) = x/(x2+1) 4. Suppose that f is a 2п-periodic differentiable function with Fouier coefficients a0, an and bn. Consider the Fourier coefficients of f ' given by a0 = 1/2п∫ f '(x) dx, an = 1/п ∫ f '(x) cos(nx) dx, bn = 1/п ∫ f '(x) sin(nx) dx, a) Show that a0 = 0. b) Using integration by parts on the formula for an and bn, find a formula for the Fourier coefficients of f ' in terms of th... click for more