The heat transfer in a semi-infinite rod can be described by the following PARTIAL differential equation:

∂u/∂t = (c^2)∂^2u/∂x^2

where t is the time, x distance from the beginning of the rod and c is the material constant. Function
u(t,x) represents the temperature at the given time t and place x. Verify that the function

u(t,x) = (e^-t)(cos x/c)
is the solution of the heat equation (i.e. it satisfies the heat equation.)

Solution Preview

Proof. Denote the partial derivative of u(t,x) with respect to t by u_t(t,x), and denote the double partial derivative of u(t,x) w.r.t x by ...

Solution Summary

This shows how to verify that a given function satisfies the heat equation for a specific situation.

Please see attachment.
4.10 Find the partialderivatives with respect to x, y, and z of the following functions:
(a) f(x,y,z) = ax^2 +bxy + cy^2
(b) g(x,y,z) = sin(axyz^2),
(c) h(x,y, z) = ax^(xy/z^2).
where a, b, and c are constants

Need help finding partialderivatives of attached problems.
Find three partialderivatives of the function r with respect to x, y, and z
1. r = uvw − u^2 − v^2 − w^2 where u = y + z, v = x + z,w = x + y
2. r = p / q + q / s + s / p where p = e^yz, q = e^xz, s = e^xy

Find (partial z)/(partial u)( and ) (partial z)/(partial v) using the chain rule. Assume the variables are restricted to domains on which the functions are defined. Your answers should be in terms of u and v.
z=arctan(x/y) x=u^2+v^2 y=u^2-v^2
partial z/partial u =
Partial z/partial v =

1. Determine the partialderivatives with respect to all of the variables in the following functions:
a.
b.
2. A company hires you as the marketing consultant to estimate the demand function for its product. You have concluded the demand function is
Where Q is the quantity demanded per capita pe

Consider the function f (x, y) given by f (x, y) = ( 1 - x^2 - y^2)^2.
a) Sketch the graph of the curve f (x, 0) for x e [-2,2].
b) Sketch the level curves for f (x, y) = c for c = 0, c = 1/4, c = 3/4 and c = 2. Also plot the level set for f (x,y) = 1.
See attachment for full question.

The voltage V in volts across a fixed resistance r in series with a variable
resistance R is V=(rE)/(R+r) where E is the source voltage. calculate the rate at
which V changes with respect to time if when E=10 volts, R=12 ohms, and r=8
ohms, the source voltage is increasing at 2 volts per minute and the variable

A. Use the chain rule to find dz/ds and dz/dt as functions of x, y, s and t
B. Find the numerical values of dz/ds and dz/dt when (s,t) = (2, -2).
Suppose z = x2 sin y, x =... (Please see the attached file for the fully formatted problem).

Solve the following PDE:
du/dt = d^2u / dx^2 (note: partialderivatives),
u(x, 0) = sin^2(x),
u(0, t) = 0, u(Pi, t) = 0,
0 < x < Pi
Repeat for the following initial condition:
du/dt (x, 0) = sin^2(x) (note:partialderivatives),
0 < x < Pi

I need to substitute a solution to the damped wave equation back into the original differential equation to get an identity. I am unsure of how to solve the partialderivatives and the steps I take mathematically going through the differential equation. A more detailed explanation is attached. Thanks!