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Consider the following partial differential equation. au, au ax? + = u ay? Identify A, B, and C in the above equation and use them to calculate the following. B2 - 4AC = -1 + u X Classify the given partial differential equation as hyperbolic, parabolic, or elliptic. O hyperbolic parabolic elliptic
Consider the below wave equation with the given conditions. au 81 Ox? u(0,1) het au 0 < x < 4, t > 0, u(4,t) = 0, 1 > 0 op u(x,0) = 0, ди at = 6x(4- x) = 384 ${1 - (-1)"} sin(npox/4), 0< x < 4. n=1 The solution to the above boundary-value problem is of the form u(x,t) = 8(n, t) sin "* n=1 Find the function g(n,1).
Part A and Part B.
Please show all work.
U AU - 5 LXTU 2. A typical microwave oven uses radiation of wavelength 12.2 cm. a. How many moles of photons of this radiation are required to raise the temperature of 500.0 mL of water from 25°C to its boiling point? wavelengthil 1.63X10 TU b. A watt (W) is a unit of power- the rate at which energy is delivered or consumed (Power (W) = Energy (J)/time (s) or 1...
Use Venn diagrams to prove or disprove the following c) AU B (An B) u (A n B)u (A n B) d) A U (B n C) (AU B) n (AU C)
1 point) Show that Φ(u, u) (Au + 2, u-u, 7u + u) parametrizes the plane 2x -y-z = 4, Then (a) Calculate Tu T,, and n(u, v). þ(D), where D = (u, u) : 0 < u < 9,0 < u < 3. (b) Find the area of S (c) Express f(x, y, z in terms of u and v and evaluate Is f(x, y,z) ds. (a) Tu n(u,v)- T, (b) Area(S)- (c) JIs f(z, y,2) ds-
1 point)...
solve the PDE
+u= at2 on 3 € (0,L), t > 0, with boundary conditions au 2x2 u(0,t) = 0, u(L, t) = 0 au and initial condition u(x,0) = f(x), at (x,0) = g(x) following the steps below. (a) Separate the variables and write differential equations for the functions (x) and h(t); pick the separation constant so that we recover a problem already studied. (b) Find the eigenfunctions and eigenvalues. (c) Write the general solution for this problem. (d)...
6. (6 points) Let u = u(,y) be a smooth function in R2 such that au u + ara aya = 0 in a simply connected region R with boundary C. Let it be the unit outward normal to R. Show that ди ds = 0, where ди Vun.
7. A is cell made by placing Zn(s) in Zn2 (aq) and Au(s) in Au (aq). Use Standard Reduction Potentials. (12 points) a. Write the half reaction for the reaction at the cathode: Calculate the voltage for the standard cell (IM Zn2* (aq) and 1 M Au?* (aq). c. Calculate the voltage for the cell after it is 999% used up. Assume 1.00 L of each solution. (Hint: Write the overall equation.)