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.

Contrasting Ordinary and Partial Derivatives. Please also explain when we should use ordinary derivatives and when we should use partial derivatives. ...

Find the partial derivatives .. Please see attachment. ... Please see the attached file. It finds the partial derivatives with respect to x, y, and z of functions. ...

Partial derivatives and price elasticity. 1. Determine the partial derivatives with respect to all of the variables in the following functions: a. b. ...

Partial Derivative and Double Integral. The problems are attached. ... 2 2 2 f (x, y) = 4x y+2y − 2 xy −4y. Solution. We find the partial derivatives as follows. ...

Partial derivatives word problem. The voltage V in volts across a fixed resistance r in series with a variable. ...Partial derivatives word problem is evaluated. ...