Repeated eigenvalues, Mathematics

Assignment Help:

It is the last case that we require to take a look at. During this section we are going to look at solutions to the system,

x?' = A x?

Here the eigenvalues are repeated eigenvalues. As we are going to be working with systems wherein A is a 2x2 matrix we will create this assumption from the start. Therefore the system will have a double eigenvalue, l.

It presents us with a problem. We need two linearly independent solutions hence we can form a general solution. Though, with a double eigenvalue we will have only one,

x? = ?h elt

Therefore, we need to come up along with a second solution. Recall this when we looked at the double root case along with the second order differential equations we ran in a same problem. In those sections we simply added a t to the solution and were capable to get a second solution. Let's notice if similar thing will work into this case as well.  We'll notice if,

x? = t elt  ?h

It will also be a solution.

To check all we want to do is plug in the system.  Keep in mind to product rule the proposed solution while you differentiate!

?h elt+ l t elt  ?h = A t elt  ?h

Here, we got two functions this time on the left side, an exponential through itself and exponential times a t. Therefore, in order for our guess to be a solution we will require to need,

A ?h=l ?h           ⇒         (A - lI) ?h= ?0

?h = ?0

The first requirement isn't a problem as this just says as l is an eigenvalue and its eigenvector is ?h. We already identified this therefore there's nothing new there. The second though is a problem. As ?h is an eigenvector we know that this can't be zero, even in order to satisfy the second condition this would have to be.

Therefore, our guess was incorrect.  The problem appears to be that there is a lone term with just an exponential into it ?hlet's see if we can't fix up our guess to accurate that. Let's try the subsequent guess.

x? = t elt  ?h +  elt?r

Here ?ris an unknown vector which we'll need to find out.

As with the first guess here we plug this in the system and notice what we get.

1741_REPEATED EIGENVALUES.png

Above is again set coefficients equal is shown.

As along with our first guess the first equation gives us nothing which we didn't already know. Now there the second equation is not a problem. Each the second equation gives us that ?r must be a solution to that equation.

It seems our second guess worked. So,

x? 2= t elt  ?h +  elt?r

It will be a solution to the system provided ?r  is a solution to;

(A - lI) ?r =  ?h

Also above solution and the first solution are linearly independent and therefore they form a fundamental set of solutions and therefore the general solution in the double eigen-value case is,

x?(t) = c1 el1t  ?h + c2 (lel2t  ?h+  elt?r)


Related Discussions:- Repeated eigenvalues

Solve the differential equation, Solve the subsequent differential equation...

Solve the subsequent differential equation and find out the interval of validity for the solution. Let's start things off along with a fairly simple illustration so we can notic

Operation of fraction, what are the formula in the operation of fraction an...

what are the formula in the operation of fraction and how will i apply the operation of fraction on word problems

Pigeonhole principle, By pigeonhole principle, show that if any five number...

By pigeonhole principle, show that if any five numbers from 1 to 8 are chosen, then two of them will add upto 9.    Answer: Let make four groups of two numbers from 1 to 8 like

Calculate values of kinetics , A reaction following first-order kinetics wa...

A reaction following first-order kinetics was studied by determining the reactant concentrations at equal time intervals. Each successive pair of concentrations, [A] o and [A] 1

Comparing fractions, I need help trying to compare 10/15 and 8/12

I need help trying to compare 10/15 and 8/12

Describe the types of triangles, Describe the Types of triangles ? Tria...

Describe the Types of triangles ? Triangles can be classified according to the lengths of the sides or the measures of the angles. 1. Naming triangles by sides An

Function of a function, Function of a Function Suppose ...

Function of a Function Suppose y is a function of z,            y = f(z) and z is a function of x,            z = g(x)

Demand Forecast, How should shoppers Stop develop its demand forecasts?

How should shoppers Stop develop its demand forecasts?

Area and surface Are, I cant figure out how to study for my math test

I cant figure out how to study for my math test

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