Distinct eigenvalues, Mathematics

Assignment Help:

It's now time to do solving systems of differential equations. We've noticed that solutions to the system,

x?' = A x?

It will be the form of,

x? = ?h elt

Here l and ?h are eigenvalues and eigenvectors of the matrix A. We will be working along with 2 x 2 systems therefore it means that we are going to be searching for two solutions, here the determinant of the matrix x?1 (t) and x?2 (t).

X =  (x?1 x?2)

They are non-zero.

We are going to start by searching the case where our two eigen-values, l1 and l2 are real and distinct. Conversely, they will be real, simple eigen-values. Recall suitably that the eigenvectors for easy eigenvalues are linearly independent. It means that the solutions we find from these will also be linearly independent. The matrix X should be nonsingular, herefore these two solutions will be a fundamental set of solutions, if the solutions are linearly independent. The general solution for this case will find be,

x?(t) = c1 el1t  ?h(1) + c2 el2t  ?h(2)

Remember that each of our illustrations will actually be broken in two illustrations. The first illustration will be solving by the system and the second illustration will be solving by sketching the phase portrait for the system. Phase portraits are not all the time taught in a differential equations course and thus we'll strip those out of the solution process hence if you haven't covered them in your class you can ignore the phase portrait illustration for the system.


Related Discussions:- Distinct eigenvalues

Geometry help, A painter leans a 10-foot ladder against the house she is to...

A painter leans a 10-foot ladder against the house she is to paint. The foot of the ladder is 3 feet from the house. How far above the ground does the ladder touch the house? Appro

Steps for integration strategy - integration techniques, Steps for Integrat...

Steps for Integration Strategy 1. Simplify the integrand, if possible This step is vital in the integration process. Several integrals can be taken from impossible or ve

Geography, How do you find the maxima or minima on a parabolic graph?

How do you find the maxima or minima on a parabolic graph?

Two circles c(o, Two circles C(O, r) and C 1 (O 1 , r 1 ) touch each other ...

Two circles C(O, r) and C 1 (O 1 , r 1 ) touch each other at P, externally or internally.  Construction: join OP and O 1 P . Proof : we know that if two circles touch each

Learning and formulating maths teaching strategies, Before going further, l...

Before going further, let us repeat an aspect of learning which is useful to keep in mind while formulating teaching strategies. A child who can add or subtract in the context of s

Twice a number increased by 11 is equal to 32 less three, Twice a number in...

Twice a number increased by 11 is equal to 32 less than three times the number. Find out the number. Let x = the number. Now translate every part of the sentence. Twice a numb

Pemdas, what is the answer using pemdas (32 divided into 4)+3

what is the answer using pemdas (32 divided into 4)+3

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd