Interpretation of ODE Question

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SUMMARY

The discussion focuses on interpreting questions related to ordinary differential equations (ODEs), specifically proving that a given function is a solution and demonstrating that it is the general solution. Participants emphasize the importance of deriving the eigen-equation for part (a) and using the definition of "general solution" for part (b). The conversation highlights that understanding the relationship between a matrix A and its eigenvectors is crucial, as well as differentiating between specific and general solutions of ODEs.

PREREQUISITES
  • Understanding of ordinary differential equations (ODEs)
  • Familiarity with eigenvalues and eigenvectors
  • Knowledge of first and second order ODEs
  • Ability to express algebraic relationships in linear algebra
NEXT STEPS
  • Study the process of deriving eigen-equations in linear systems
  • Learn the definitions and distinctions between specific and general solutions of ODEs
  • Explore methods for solving first and second order ODEs
  • Research the role of eigenvectors in the context of differential equations
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Students and educators in mathematics, particularly those focusing on differential equations and linear algebra, as well as professionals working with systems of ODEs.

Ry122
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Hi

Could someone please help me to understand what questions a) and b) here are asking for?

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Pretty much means what it says:
It wants you to prove that (1) is a solution to the DE given, and then prove that (1) is actually the general solution.
 
Okay well for a) I'm guessing you would do this by deriving the eigen-equation.

But for b) how would you show that it's not just 'a' solution but is the general solution?
 
Ry122 said:
Okay well for a) I'm guessing you would do this by deriving the eigen-equation.
There is no need to derive anything - the equations are given to you. Though you do need to be able to express the relationship between A and it's eigenvectors algebraically.

But for b) how would you show that it's not just 'a' solution but is the general solution?
Use the definition of "general solution".
What is the difference between a specific solution and the general solution of any DE?

All this is stuff you should have covered way back when you 1st learned about 1st and 2nd order ODE's - before you ad to deal with systems of ODEs like above. It is just the same.
 

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