Partial Derivatives ............. help Please 10pts?

[匿名]

2011-10-10 6:59 pm

1.) Show that u(x,t) = sin(nx)e^(-n^(2)t) satisfies the heat equation for any constant n:

df/dt = d^2f/dx^2.

2.) Show that the functions is harmonic:

u(x,y) = tan^-1 y/x.

回答 (1)

2011-10-10 7:19 pm

✔ 最佳答案

1) ∂u/∂t = -n² sin(nx) e^(-n²t)

∂u/∂x = n cos(nx) e^(-n²t)
∂²u/∂x² = -n² sin(nx) e^(-n²t)

Hence, ∂u/∂t = ∂²u/∂x².
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2) We need to show that u satisfies Laplace's Equation ∂²u/∂x² + ∂²u/∂y² = 0.

∂u/∂x = (1/(1 + (y/x)²) * (-y/x²) = -y/(x² + y²).
==> ∂²u/∂x² = 2xy/(x² + y²)²

∂u/∂y = (1/(1 + (y/x)²) * (1/x) = x/(x² + y²).
==> ∂²u/∂y² = -2xy/(x² + y²)².

So, ∂²u/∂x² + ∂²u/∂y² = 0, as required.
==> u is harmonic.

I hope this helps!

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