8E: Excercise

Exercise $\PageIndex{1}$: Heat Equation

1. Let $u(x,t)= t^{-1/2} e^{-x^2/4t}.$ Find $\frac{\partial ^2u}{\partial x^2}-\frac{\partial u}{\partial t}.$
2. Let $u(x,y,t)= t^{-1} e^{-(x^2+y^2)/4t}.$ Find $\frac{\partial ^2u}{\partial x^2}+\frac{\partial ^2u}{\partial y^2}-\frac{\partial u}{\partial t}.$

Exercise $\PageIndex{2}$: Laplace equation

Show that function $z(x, y) = 2e^{\pi x} sin(\pi y)$ satisfies the two dimensional Laplace equation: $$\frac{\partial ^2z}{\partial x^2}+\frac{\partial ^2z}{\partial y^2}=0.$$