8E: Excercise

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Exercise \(\PageIndex{1}\): Heat Equation

Let \( u(x,t)= t^{-1/2} e^{-x^2/4t}.\) Find \(\frac{\partial ^2u}{\partial x^2}-\frac{\partial u}{\partial t}.\)
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: \[\frac{\partial ^2z}{\partial x^2}+\frac{\partial ^2z}{\partial y^2}=0.\]

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