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

Find out the x-y coordinates of the points - tangents, Find out the x-y coo...

Find out the x-y coordinates of the points in which the following parametric equations will have horizontal or vertical tangents. x = t 3 - 3t        y = 3t 2 - 9 Solut

Determine the determinant of matrix, Example Determinant:   Determine ...

Example Determinant:   Determine the determinant of each of the following matrices. Solution : For the 2 x 2 there isn't much to perform other than to plug this in

probability problems, A school principal is looking at the combinations of...

A school principal is looking at the combinations of subjects students are studying. He learns that the probability that a student is studying Chemistry is 0.5 and that the prob

What is this distance expressed in standard notation, The distance from the...

The distance from the sun to the earth is approximately 9.3 × 10 7 miles. What is this distance expressed in standard notation? In order to convert this number to standard not

Find the area of the shaded region of square, In the adjoining figure, ABCD...

In the adjoining figure, ABCD is a square of side 6cm.  Find the area of the shaded region. Ans:    From P draw PQ ⊥ AB AQ = QB = 3cm (Ans: 34.428 sq cm) Join PB

Probability questions, The probability that a person will get an electric c...

The probability that a person will get an electric contract is 2/5 and the contract that he will not get plumbing  contract is 4/7.If the probability of getting at least one contra

Evaluate the area of the region, Evaluate the area of the region. a...

Evaluate the area of the region. a. 478 units 2 b. 578 units 2 c. 528 units 2 d. 428 units 2   b. Refer to the diagram to evaluate the area of the shaded

Partial fractions - integration techniques, Partial Fractions - Integration...

Partial Fractions - Integration techniques In this part we are going to take a look at integrals of rational expressions of polynomials and again let's start this section out w

General solution to a differential equation, The general solution to a diff...

The general solution to a differential equation is the most common form which the solution can take and does not take any initial conditions in account. Illustration 5: y(t) =

Example of cartesian coordinate graph, Example of Cartesian coordinate Grap...

Example of Cartesian coordinate Graph: Example:   The temperature of water flowing in a high pressure line was measured at regular intervals.  Plot the subsequent recorded da

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