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