Find the deflection u(x, y, t) of the square membrane of side π and c²=1 for initial velocity 0 and initial deflection 0.01 sin x sin y

Find the deflection u(x, y, t) of the square membrane of side π and c²=1 for inital velocity 0 and inital deflection 0.1 sin 2x sin 4y

Find the deflection u(x, t) of the string of length L=π and c²=1 for zero initial displacement and "triangular" initial velocity u_t(x, 0)=0.01x if 0≤x≤½π, u_t(x, 0)=0.01(π-x) if ½π≤x≤π. (Initial conditions with u_t(x, 0)≠0 are hard to realize experimentally.)

Find all solutions of each of the following linear congruences in two variables. a) 2x + 3y = 1 (mod 7), b) 2x + 4y = 6 (mod 8), c) 6x + 3y = 0 (mod 9), d) 10x + 5y = 9 (mod 15).

Solve the following ODEs, showing the details of your work. (D³-9D²+27D-27l)y=27sin3x

Find u(x, t) for the string of length L=1 and c²=1 when the initial velocity is zero and the initial deflection with small k(say, 0.01) is as follows. Sketch or graph u(x,t). kx(1-x)

What is an example of heat transfer by convection?

The faces of the thin square plate with side a=24 are perfectly insulated. The upper side is kept at 25°C and the other sides are kept at 0°C. Find the steady-state temperature u(x, y) in the plate.

When crossing an intersection, a motorcyclist encounters the slight bump or crown caused by the intersecting road. If the crest of the bump has a radius of curvature ρ = 50 ft, determine the maximum constant speed at which he can travel without leaving the surface of the road. Neglect the size of the motorcycle and rider in the calculation. The rider and his motorcycle have a total weight of 450 lb.

A paratrooper and his 8-m-diameter parachute weigh 950 N. Taking the average air density to be $1.2 kg/m^3,$ determine the terminal velocity of the paratrooper.

Find the electrostatic potential in the semidisk r<1, 0<θ<π which equals 110θ(π-θ) on the semicircle r=1 and 0 on the segment -1<x<1.

Find the temperature with $L=\pi$. $c=1$, and $f(x)=r$

In a science museum, you may have seen a Foucault pendulum, which is used to demonstrate the rotation of the earth. In one museum's pendulum, the $110 \mathrm{~kg}$ bob swings from a $15.8$-m-long cable with an amplitude of $5.0^{\circ}$.
a. What is the period of this pendulum?
b. What is the bob's maximum speed?
c. What is the pendulum's maximum kinetic energy?
d. When the bob is at its maximum displacement, how much higher is it than when it is at its equilibrium position?

Find the temperature u(x,t) in a bar of silver of length 10 cm and constant cross section of area 1 cm² (density 10.6 g/cm³, thermal conductivity 1.04 cal/(cm sec °C), specific heat 0.056 cal/(g °C) that is perfectly insulated laterally, with ends kept at temperatures 0°C and initial temperature f(x)°C, where f(x)=4-0.8|x-5|