Distinct eigenvalues –system solving, Mathematics

Assignment Help:

DISTINCT EIGENVALUES -SYSTEM SOLVING:

Example

Solve the following IVP.

1197_DISTINCT EIGENVALUES –SYSTEM SOLVING.png

Solution:

Therefore, the first thing that we must to do that is, get the eigenvalues for the matrix.

1334_DISTINCT EIGENVALUES –SYSTEM SOLVING1.png

= l2 -3l - 4

= (l+ 1) (l- 4)                                     ⇒                     l1 = -1,  l2 = 4

Here let's get the eigenvectors for each of these.

For l1 = -1,

We have to solve:

2433_DISTINCT EIGENVALUES –SYSTEM SOLVING2.png

= 2h1 + 2h2                              ⇒                                 h1 = - h2

The eigenvector for this case is,

2482_DISTINCT EIGENVALUES –SYSTEM SOLVING3.png

= h2= 1,

For l2 = 4,

We have to solve:

1479_DISTINCT EIGENVALUES –SYSTEM SOLVING4.png

⇒ = -3h1 + 2h2 = 0                              ⇒                                 h1 =( 2/3) h2

The eigenvector for this case is,

1101_DISTINCT EIGENVALUES –SYSTEM SOLVING5.png

= h2 = 3

So the general solution is after that,

1221_DISTINCT EIGENVALUES –SYSTEM SOLVING6.png

Now, we have to get the constants. To do that we simply have to apply the initial conditions

530_DISTINCT EIGENVALUES –SYSTEM SOLVING7.png

All we required to do now is multiply the constants with and we after that get two equations that is one for each row which we can solve for the constants. It gives,

-c1 + 2c2 = 0;

c1 + 3c2 = -4;

By solving both equations we get:

c1 = -(8/5) and c2 = (-4/5)

The solution is subsequently,

 

1221_DISTINCT EIGENVALUES –SYSTEM SOLVING8.png


Related Discussions:- Distinct eigenvalues –system solving

Triple integrals, Consider a circular disc of radius 1 and thickness 1 whic...

Consider a circular disc of radius 1 and thickness 1 which has a uniform density 10 ?(x, y, z) = 1. (a) Find the moment of inertia of this disc about its central axis (that is, the

An even function, Assume that   i)  Determine all the roots of f...

Assume that   i)  Determine all the roots of f(x) = 0. ii)  Determine the value of k that makes h continuous at x = 3. iii)  Using the value of k found in (ii), sh

2 step equations, What is a two step equation that equals 8 ?

What is a two step equation that equals 8 ?

Algebra2;, log6 X + log6 (x-5) = 1

log6 X + log6 (x-5) = 1

Decimals, how to make 2.3 into a fraction?

how to make 2.3 into a fraction?

20 MARK QUESTION, Let E; F be 2 points in the plane, EF has length 1, and l...

Let E; F be 2 points in the plane, EF has length 1, and let N be a continuous curve from E to F. A chord of N is a straight line joining 2 points on N. Prove if 0 Prove that N ha

Method of disks or the method of rings, Method of disks or the method of ri...

Method of disks or the method of rings One of the simple methods for getting the cross-sectional area is to cut the object perpendicular to the axis of rotation.  Carrying out

Prisoners dilemma, Prisoners Dilemma This is a type of non-zero sum gam...

Prisoners Dilemma This is a type of non-zero sum game and derives its name from the given story: The district attorney has two bank robbers in separate cells and offers them

Estimate the probability, The following (artificial) data record the length...

The following (artificial) data record the length of stay (in days) spent on a psychiatric ward for 28 consecutive patients who have been sectioned under the mental health act, cla

Numerical integration - simpson rule, (1)Derive, algebraically, the 2nd ord...

(1)Derive, algebraically, the 2nd order (Simpson's Rule) integration formula using 3 equally spaced sample points, f 0 ,f 1 ,f 2 with an increment of h. (2) Using software such

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