Calculus: Verify Thick Walled Cylinder Equations

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Homework Help Overview

The discussion revolves around verifying Lame's exact solutions for a thick-walled cylinder subjected to internal pressure, specifically through a system of ordinary differential equations. Participants are exploring the integration of these equations to derive expressions for displacement and radial stress.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants describe attempts to substitute known values into the equations but encounter difficulties, particularly with the relationships between variables and integration constants. There are questions regarding the integration process and boundary conditions.

Discussion Status

The discussion is ongoing, with participants sharing their attempts and expressing confusion about specific steps in the integration process. Some guidance has been offered regarding the need to show work in substitutions, but no consensus has been reached on the approach to take.

Contextual Notes

There are mentions of boundary conditions related to radial stress and the nature of the equations being double integrals, which may be influencing the participants' understanding and progress.

Ketav
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Homework Statement



I have a system of two ordinary differential equations as shown below. I have to prove that the Lame's exact solutions for a thick walled cylinder loaded by internal pressure satisfies the equations.

The next step is to integrate the equations to obtain an equation for U and radial stress

upload_2016-11-8_17-15-31.png

Homework Equations


How should i go about the solution of this problem

The Attempt at a Solution


I have tried to find solutions for this but get stuck after the substitution of all given info[/B]
 
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I have tried to substitute the equations but for the hoop stress I obtain the relationship between b and r as one.
If anyone has any suggestions, I will be very thankful to you.
I have tried solving the differential equations however, i get to a point where I get d2u/dr2 + i/r du/dr + u/r=0 and get stuck with integration constants
 
Ketav said:
I have tried to substitute the equations but for the hoop stress I obtain the relationship between b and r as one.
If anyone has any suggestions, I will be very thankful to you.
I have tried solving the differential equations however, i get to a point where I get d2u/dr2 + i/r du/dr + u/r=0 and get stuck with integration constants
This is really a math problem. I'm going to move it to the math homework forum. Please show us your work in doing the substitutions.
 
So I've done this,

now I am confused about the integration. its a double integral but the boundary conditions are for r and sigma r
upload_2016-11-15_1-0-33.png
 

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