Undergrad Non-homogeneous systems with repeated eigenvalues

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SUMMARY

Non-homogeneous systems with repeated eigenvalues can be solved using methods such as variation of parameters and undetermined coefficients. The fundamental matrix, defined by the Cauchy problem ##\dot X(t)=A(t)X,\quad X(0)=I##, plays a crucial role in finding the solution. The solution to the initial inhomogeneous system is expressed as ##x(t)=X(t)\hat x+\int_0^t X(t)X^{-1}(s)b(s)ds##, where ##b(t)## is the non-homogeneous term. This confirms that the same techniques apply regardless of the presence of repeated eigenvalues.

PREREQUISITES
  • Understanding of non-homogeneous differential equations
  • Familiarity with fundamental matrices in linear algebra
  • Knowledge of variation of parameters and undetermined coefficients methods
  • Basic concepts of eigenvalues and eigenvectors
NEXT STEPS
  • Study the derivation and properties of fundamental matrices in differential equations
  • Explore the method of variation of parameters in detail
  • Learn about the application of undetermined coefficients for solving differential equations
  • Investigate the implications of repeated eigenvalues in system stability
USEFUL FOR

Mathematicians, engineers, and students studying differential equations, particularly those focusing on systems with repeated eigenvalues and their solutions.

faradayscat
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Quick question, can you solve non-homogeneous systems with repeated eigenvalues the same ways? i.e. variation of parameters, undetermined coefficients, etc... would the fundamental matrix contain the solution with the generalized eigenvalue?

Thanks!
 
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faradayscat said:
Quick question, can you solve non-homogeneous systems with repeated eigenvalues the same ways?
Sure you can. Actually this question is not about eigenvalues

Consider a system ##\dot x=A(t)x+b(t),\quad x\in\mathbb{R}^m##, here ##A(t)## is a square matrix with coefficients depending on t.
The fundamental matrix ##X(t)## is defined by means of the Cauchy problem ##\dot X(t)=A(t)X,\quad X(0)=I##. Then the solution ##x(t),\quad x(0)=\hat x## to the initial inhomogeneous system is given by the formula
##x(t)=X(t)\hat x+\int_0^t X(t)X^{-1}(s)b(s)ds##
 

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