check-that-the-function-u-x-t-exp-n-2-2-pi-2-L-2-t-sin-npix-L-n-1-2-satisfy-the-heat-equation-heat-equation-2-2-u-x-2-u-t-0-lt-x-lt-L-

Question Number 168977 by MikeH last updated on 23/Apr/22

check that the function

u (x, t) = \exp {- \frac{n^{2} α^{2} π^{2}}{L^{2}} t} \sin \frac{n π x}{L}

n = 1, 2, \dots satisfy the heat equation

heat equation

α^{2} \frac{\partial^{2} u}{\partial x^{2}} = \frac{\partial u}{\partial t}, 0 < x < L