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

Estimate the loss to the bank - restructure the loan, Your bank has a loan ...

Your bank has a loan outstanding with a current balance of $1,000,000 that is payable in quarterly equal instalments of $49,924.  This loan has another 6 years to maturity.  The bo

Formulas for the volume of this solid, Formulas for the volume of this soli...

Formulas for the volume of this solid V = ∫ b a A ( x) dx          V = ∫ d c A ( y ) dy where, A ( x ) & A ( y ) is the cross-sectional area of the solid. There are seve

Solution set of equation, The complete set of all solutions is called as th...

The complete set of all solutions is called as the solution set for the equation or inequality.  There is also some formal notation for solution sets.  We have to still acknowledge

Area between two curves, Area between Two Curves We'll start with the ...

Area between Two Curves We'll start with the formula for finding the area among y = f(x) and y = g(x) on the interval [a,b].  We will also suppose that f(x) ≥ g(x) on [a,b].

Intermediate value theorem, Intermediate Value Theorem Suppose that f(x...

Intermediate Value Theorem Suppose that f(x) is continuous on [a, b] and allow M be any number among f(a) and f(b).   There then exists a number c such that, 1. a 2. f (

Solve the linear programming problem using simple method, Solve the followi...

Solve the following Linear Programming Problem using Simple method. Maximize Z= 3x 1 + 2X 2 Subject to the constraints:                  X 1 + X 2 ≤ 4

Comparison test or limit comparison test, Comparison Test or Limit Comparis...

Comparison Test or Limit Comparison Test In the preceding section we saw how to relate a series to an improper integral to find out the convergence of a series.  When the inte

Cartesian Coordinates, In the view below of the robot type of Cartesian Coo...

In the view below of the robot type of Cartesian Coordinates, is not the "Z" and "Y" coordinates reversed? http://www.expertsmind.com/topic/robot-types/cartesian-coordinates-91038

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