**1. The Heat Equation**The heat equation is an equation involving the partial derivatives of a function $ u(x, t) $:\[\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}\]Show that $ u(x, t) = \cos(nx) e^{-n^2 t} $ satisfies the heat equation for any constant $ n $.---**2. Horizontal Tangent Plane**For which points $ (x, y) $ is the tangent plane to $ z = \sin(x) \sin(y) $ horizontal?

The heat equation is ∂u∂t=∂2u∂x2. To show: ux, t=cosnxe-n2t satisfies the heat equation.…

Answered: 1. The heat equation is an equation…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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**1. The Heat Equation**

The heat equation is an equation involving the partial derivatives of a function $ u(x, t) $:

\[
\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}
\]

Show that $ u(x, t) = \cos(nx) e^{-n^2 t} $ satisfies the heat equation for any constant $ n $.

---

**2. Horizontal Tangent Plane**

For which points $ (x, y) $ is the tangent plane to $ z = \sin(x) \sin(y) $ horizontal?

Expert Solution

Solution 1:

The heat equation is $\frac{\partial u}{\partial t} = \frac{\partial^{2} u}{\partial x^{2}}$ .

To show: $u (x, t) = \cos (n x) e^{- n^{2} t}$ satisfies the heat equation.

Differentiate u partially with respect to t:

$\begin{array}{rcl} \frac{\partial u}{\partial t} & = & \cos (n x) \frac{\partial}{\partial t} e^{- n^{2} t} \\ = & \cos (n x) e^{- n^{2} t} \frac{\partial}{\partial t} (- n^{2} t) \\ = & \cos (n x) e^{- n^{2} t} (- n^{2}) \\ = & - n^{2} \cos (n x) e^{- n^{2} t} \end{array}$

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