D’Alembert’s reduction principle

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SUMMARY

The discussion centers on D’Alembert’s reduction principle in the context of an ordinary differential equation (ODE) system. Participants clarify that FS refers to the fundamental system, and emphasize that the provided equations lack derivatives, indicating they do not constitute an ODE system. The correct formulation of the equations is highlighted, specifically that the second equation should be dx2(t)/dt = x1(t)/t^2 + 2x2(t)/t. The known particular solution (t^2, -t) is identified as a solution to an algebraic equations system rather than an ODE.

PREREQUISITES
  • Understanding of ordinary differential equations (ODEs)
  • Familiarity with D’Alembert’s reduction principle
  • Knowledge of fundamental systems in differential equations
  • Basic calculus, specifically differentiation
NEXT STEPS
  • Study the application of D’Alembert’s reduction principle in solving ODEs
  • Learn how to identify and formulate ordinary differential equations
  • Explore the concept of fundamental systems in differential equations
  • Review techniques for solving algebraic equations systems
USEFUL FOR

Students and professionals in mathematics, particularly those studying differential equations, as well as educators seeking to clarify concepts related to ODE systems and algebraic equations.

mathrocks
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Hey guys, I'm stuck on this problem! I have no idea how to even go about solving it. I tried searching the net for D'Alembert's principle but nothing was helpful. Any suggestions on how to go about solving it will be much appreciated!

Find with the aid of D’Alembert’s reduction principle the FS of the ODE system.

x1(t)= x1(t)/t - x2(t)

x2(t)= x1(t)/t2+ 2x2(t)/2

for which a particular solution x(t)=(t2, -t) is known.
 
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Is this an ODE System or algebric equations System?

What is FS (sorry I don't know :P)?
 
elibj123 said:
Is this an ODE System or algebric equations System?

What is FS (sorry I don't know :P)?


this is an ODE. FS stands for fundamental system.
 
There are no derivatives, therefore this is not a ODE. Secondly (t^2,-t) is not a solution.
Missing derivatives, and the second line should be,
x_2' = x_2/t^2 + 2 x_2/t
 
Last edited:
that's the point,
(t^2,-t) is the solution of the algebric equations (as he written them)
so I'm a bit confused
 
Mathrocks, do you mean
dx1(t)/dt= x1(t)/t - x2(t)

dx2(t)/dt= x1(t)/t2+ 2x2(t)/2
 

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