Find the solution of the partial differential equation-dxazay?- 8= 0by variable separable methodMaximum file size: 100MB, maximum number of files: 1FilesYou can drag and drop files here to add them.Accepted file typesDocument files .doc.docx.epub.gdoc.odt .oth .ott pdf.rtfImage files .ai .bmp .gdraw .gif .ico .jpe .jpeg jpg .pct .pic .pict .png .svg .svgz .tif .tiffCauabu Differential Eguation

Consider the given: ∂2z∂x2−8∂2z∂y2=0. Assume z(x,y)=X(x)Y(y). Therefore, ∂2z∂x2=X''Y and…

Answered: Find the solution of the partial…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.3: The Natural Exponential Function

Problem 44E

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Question

Find the solution of the partial differential equation-
dx
az
ay?
- 8
= 0
by variable separable method
Maximum file size: 100MB, maximum number of files: 1
Files
You can drag and drop files here to add them.
Accepted file types
Document files .doc.docx.epub.gdoc.odt .oth .ott pdf.rtf
Image files .ai .bmp .gdraw .gif .ico .jpe .jpeg jpg .pct .pic .pict .png .svg .svgz .tif .tiff
Cauabu Differential Eguation

Expert Solution

Step 1

Consider the given:

$\frac{\partial^{2} z}{\partial x^{2}} - 8 \frac{\partial^{2} z}{\partial y^{2}} = 0$ .

Assume $z (x, y) = X (x) Y (y)$ .

Therefore, $\frac{\partial^{2} z}{\partial x^{2}} = X^{''} Y$ and $\frac{\partial^{2} z}{\partial y^{2}} = X Y^{''}$ .

So, our partial differential equation (PDE) becomes

$\begin{array}{l} X^{''} Y - 8 Y^{''} X = 0 \\ X^{''} Y = 8 Y^{''} X \\ \frac{X^{''}}{8 X} = \frac{Y^{''}}{Y} = k (Separate Variables) \end{array}$

Hence, $X^{''} - 8 k X = 0$ and $Y^{''} - k Y = 0$ .

Step 2

When $k = 0$ ,

Then equation becomes $X " = 0, Y^{''} = 0$ .

Implies

$X (x) = c_{1} x + c_{2}, Y (y) = c_{3} y + c_{4}$

Therefore,

$\begin{array}{l} z (x, y) = X (x) Y (y) \\ = (c_{1} x + c_{2}) (c_{3} y + c_{4}) \end{array}$

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Find the solution of the partial differential equation- -8 0: ax2 ay? by variable separable method