Solving a 2nd order non-linear DE by dimensional analysis/observation

AI Thread Summary
The discussion centers on solving a second-order non-linear differential equation from Morin's Classical Mechanics, specifically the form r = Agt^2. Participants seek clarification on why g must be a variable in r and the reasoning behind A being a dimensionless constant. Dimensional analysis is highlighted as a method for ensuring dimensional consistency in equations, leading to the conclusion that A can be expressed as a combination of g, t, and r. The dimensions of g are established as LT^-2, while the relationship between the variables is explored to eliminate dimensions. Understanding these concepts is crucial for solving the differential equation effectively.
phantomvommand
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Homework Statement
See picture below
Relevant Equations
Dimensional analysis
Solving DE
Screenshot 2021-10-01 at 4.08.55 AM.png

The top most 2nd order non-linear DE is the one that has to be solved. Below is the solution. This problem is from Morin's Classical Mechanics.
May I know how he could guess that r = Agt^2?
Firstly, why must g tilda be a variable within r? I do not understand what he meant by 'parameter'.
Secondly, how was it deduced that A must be a constant with no units? Aren't there many constants with units?

Thanks for all your help.
 
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What are the units of g?
 
Chestermiller said:
What are the units of g?
ms-2?
 
phantomvommand said:
Homework Statement:: See picture below
Relevant Equations:: Dimensional analysis
Solving DE

View attachment 289987
The top most 2nd order non-linear DE is the one that has to be solved. Below is the solution. This problem is from Morin's Classical Mechanics.
May I know how he could guess that r = Agt^2?
Firstly, why must g tilda be a variable within r? I do not understand what he meant by 'parameter'.
Secondly, how was it deduced that A must be a constant with no units? Aren't there many constants with units?

Thanks for all your help.
Have you read up on dimensional analysis?
The notion is that for dimensional consistency any equation can be written as some functional combination of dimensionless terms.

In the present case we believe there is an equation relating ##\tilde g, t, r##, and no other variables of dimension. So it can only involve those variables in dimensionless combinations.
Writing square brackets to mean "dimension of":
##[\tilde g]=LT^{-2}##
##[t]=T##
##[r]=L##
How can these be combined to eliminate the dimensions? Only as ##\frac{\tilde g t^2}r##. So we can write ##A=\frac{\tilde g t^2}r##, where A is dimensionless.
 
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