Final answer:
The equilibrium solution of the heat equation uₜ=4uₓₓ is u(x)=c, where c is any constant.
Step-by-step explanation:
The heat equation is a partial differential equation that describes the distribution of temperature in a given region over time. The equilibrium solutions are those where the temperature remains constant over time. To find the equilibrium solution of the heat equation uₜ=4uₓₓ, we set the time derivative uₜ equal to 0, since there is no change over time in the equilibrium solution. So, we have: 0=4uₓₓ.
To solve this equation, we can assume a solution of the form u(x)=c, where c is a constant. Substituting this into the equation gives us: 0=4cₓₓ. Since c is a constant, its derivative is zero, so the equation simplifies to: 0=0. Thus, any constant value of c is a solution to the equation. Therefore, the equilibrium solution of the heat equation uₜ=4uₓₓ is u(x)=c, where c is any constant.