4. (a) Verify u(r, t) = e-r ,x € R', t > 0 satisfies the following partial differential equation:%3D2ntUt – Upa = 0,xE R',t > 0.(b) Find the following integral| u(x, t)dz.

Answered: 4. (a) Verify u(x, t) = e-,rE R',t> 0…

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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4. (a) Verify u(r, t) = e-r ,x € R', t > 0 satisfies the following partial differential equation:
%3D
2nt
Ut – Upa = 0,
xE R',t > 0.
(b) Find the following integral
| u(x, t)dz.

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Step 1

A scalar function $f (x, t)$ is a rule that is used to associate a number or quantity with a particular point $(x, t)$ in a plane. If $y = f (x, t)$ , then the independent variable x and t are called the input of the function, and the dependent variable y is called the output of the function.

The partial derivative of a multivariable function with respect to a variable is the derivative of a function with respect to that particular variable, by treating all other variables as constants. For example, if $f (x, y, z) = x^{2} y^{2} z^{2}$ is a multivariable function, then the partial derivative of f with respect to x is $\frac{\partial f}{\partial x} = 2 x y^{2} z^{2}$ (since the derivative of $x^{2}$ with respect to x is $2 x$ and $y^{2} z^{2}$ is treated as constant while calculating the partial derivative).

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