Solving DE with Constant: Homogenous & Non-Homogenous

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SUMMARY

The discussion focuses on solving differential equations (DEs) that include a constant term, specifically examining the equation y'' + 2y' + c = 0, where c is a constant. It clarifies that such equations do not fit neatly into the categories of homogeneous or non-homogeneous DEs. The solution method for these equations involves recognizing them as non-homogeneous when a constant is present, leading to a general solution format that includes exponential decay and constants of integration. An example provided is the equation AEu'' + f = 0, where f is a constant, demonstrating the application of these principles.

PREREQUISITES
  • Understanding of differential equations, particularly homogeneous and non-homogeneous types.
  • Familiarity with solving second-order linear DEs.
  • Knowledge of the method of undetermined coefficients for non-homogeneous DEs.
  • Basic grasp of boundary value problems in applied mathematics.
NEXT STEPS
  • Research the method of undetermined coefficients for solving non-homogeneous differential equations.
  • Study the characteristics of second-order linear differential equations with constant coefficients.
  • Explore applications of differential equations in structural engineering, particularly in analyzing deflection under loads.
  • Learn about the Laplace transform method for solving differential equations with constant terms.
USEFUL FOR

Mathematicians, engineering students, and professionals dealing with differential equations in physics and engineering applications, particularly those focused on structural analysis and dynamic systems.

phiby
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My math is a little rusty - so please bear with me if this is a stupid question.

I know how to solve both homogenous and non-homogenous DEs

However, I am not sure where a DE with a constant falls & how to solve it.

For eg.

y'' + 2y' + c = 0
(c is a constant).

You cannot convert this into an algebraic equation in r like you do for regular homogenous DEs. So what's the method for solving this?

If this is a different category of DEs (i.e. neither homo nor non-homo), then even giving me the name of this type of DE is good enough - I can google and find the method.

An example of this type of DE is a bar loaded with a uniformly distributed load of f.
The DE is

AEu'' + f = 0

u -> deflection.

f is a constant.
 
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You can solve the equation y'' + 2y' = -c. This is a non homogenous equation where the non-homogenous part is a constant. So the general solution is

-((c x)/2) - 1/2 exp(-2 x) C[1] + C[2]

where C[1] and C[2] are constants.
 

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