Matrix Differential Equation with Generalized Eigenvectors

Click For Summary
SUMMARY

The discussion centers on solving a matrix differential equation represented by the system x' = | 0 1 | * x with initial conditions x(0) = | 2 | and | 3 |. The eigenvalues identified are both 5, with the corresponding eigenvector being [1, 5]. To find the generalized eigenvector, the user applies the formula (A - 5I)v_2 = v_1, resulting in v_2 = [0, 1]. The user seeks guidance on the appropriate form for the solution to the differential equation.

PREREQUISITES
  • Understanding of matrix differential equations
  • Knowledge of eigenvalues and eigenvectors
  • Familiarity with generalized eigenvectors
  • Basic proficiency in linear algebra concepts
NEXT STEPS
  • Study the method for solving matrix differential equations using eigenvalues and eigenvectors
  • Learn about the construction of solutions involving generalized eigenvectors
  • Explore the application of the Jordan form in solving differential equations
  • Review examples of initial value problems in linear systems
USEFUL FOR

Students preparing for exams in differential equations, mathematicians focusing on linear algebra, and educators teaching matrix theory and its applications.

patchwerk
Messages
3
Reaction score
0
Hey guys, need some quick help before an exam

I have a differential eqn.

x' = | 0 1 | *x , and initial conditions x(0) = |2|
| -25 10 | |3|

I find that there are two eigenvalues 5, and 5

The corresponding eigenvector to 5 is [1 5] (vertical)

So i need to find a generalized eigenvector,

I do so in the form

(A - 5lambda)v_2 = v_1

I then find that v_2 = [0 1 ] (vertically)

I don't know what form my solution should now be in

Please help, I have an exam at 7,

Thanks,

Evan
 
Physics news on Phys.org
the matrix is R1: 0 1 and R2: -25 10
and the IC is x(0) = R1:2 R2:13

sorry, not sure hwo to enter matrices
 
patchwerk said:
the matrix is R1: 0 1 and R2: -25 10
and the IC is x(0) = R1:2 R2:13

sorry, not sure hwo to enter matrices

R2: is 3 sorry!
 

Similar threads

Undergrad Fundamental matrix of a second order 2x2 system of ODEs
  • · Replies 2 ·
Replies
2
Views
4K
MHB Differential equation with a matrix
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Generalized eigenvectors and differential equations
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Can one find a matrix that's 'unique' to a collection of eigenvectors?
  • · Replies 33 ·
2
Replies
33
Views
3K
Undergrad General solution vs particular solution
  • · Replies 52 ·
2
Replies
52
Views
9K
MHB Finding Eigenvalues, Eigenvectors, [3]
  • · Replies 17 ·
Replies
17
Views
4K
Graduate Differential equation and Appell polynomials
  • · Replies 1 ·
Replies
1
Views
4K
Undergrad On vibrating string and differentiating infinite sum
  • · Replies 7 ·
Replies
7
Views
3K
Fundamental matrix for complex linear DE system
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Unravelling Structure of a Symmetric Matrix
  • · Replies 5 ·
Replies
5
Views
3K