Second order DE quick question

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SUMMARY

The discussion clarifies the origin of constants A and B in the general solution of a second-order differential equation (ODE) with distinct roots. Specifically, for the equation y = e^(mx), the general solution is expressed as y = Ae^(mx) + Be^(m_1x), where A and B represent arbitrary constants arising from the linear nature of the ODE. This linearity allows for the inclusion of multiple independent solutions, confirming that if e^(alpha x) is a solution, then any scalar multiple A * e^(alpha x) is also a valid solution.

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  • Understanding of second-order differential equations
  • Familiarity with the concept of linear independence in solutions
  • Knowledge of the auxiliary equation and its roots
  • Basic principles of the differentiation operator
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Students and professionals in mathematics, particularly those studying differential equations, as well as educators seeking to explain the concepts of linearity and solution independence in ODEs.

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I don't understand where the A and B come from,

if y = e^(mx), would the general solutions be y = Ae^(mx) + Be^(m_1x) assuming there are two distinct roots of the auxiliary equation? If anyone could clear this up, thanks.
 
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A and B are the 'freedom' of the ODE, that is the maximum number of arbitrary constants which multiply the linear independent solutions. This 'freedom' is granted by the linear character of the equation (which comes from the linear characted of the differentiation operator).

In other words, if e^(alpha x) is a solution of the ODE, so is any A times e^(alpha x). The B comes from the second independent solution.
 

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