• :00Days
  • :00Hours
  • :00Mins
  • 00Seconds
A new era for learning is coming soonSign up for free
Log In Start studying!

Select your language

Suggested languages for you:
Answers without the blur. Sign up and see all textbooks for free! Illustration


Linear Algebra With Applications
Found in: Page 426
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

Answers without the blur.

Just sign up for free and you're in.


Short Answer

Question: Consider the system dxdt=[01-10]Ax whereA=[0100] . Sketch a direction field for Base on your sketch, describe the trajectories geometrically. Can you find the solution analytically?


The solution of the system is xt=c1+c2+c2tc2 .

See the step by step solution

Step by Step Solution

Step 1: Determine the Eigen values of the matrix.

Consider the equation dxdt=Axwith A=0100.

Assume is an Eigen value of the matrix 0100 implies .

Substitute the values 0100for A androle="math" localid="1660643543411" 1001 for in the equation as followsA-λI=0.


Simplify the equation0100-λ1001=0 as follows.


Therefore, the Eigen values of A are λ=0.

Step 2: Determine the Eigen vector corresponding to the Eigen value λ=0

Assume v1=x1y1 and v2=x2y2 are Eigen vector corresponding toλ=0,0 implies A-λ1Iv1=0 and A-λ2Iv2=0 .

Substitute the values 0100for A,0 for role="math" localid="1660644995182" λ1x1y1, for v1 and for in the equationA-λ1Iv1=0 as follows.


As is chosen to be arbitrary, assume x1 = 1 implies x1y1=10

Therefore, the Eigen vector corresponding to role="math" localid="1660645072341" λ1=0 is x1y1=10 .

Substitute the values0100 for A, 0 for , for and1001 for in the equation as follows.

As is chosen to be arbitrary, assume implies 0100

Therefore, the Eigen vector corresponding to λ2=0 is x2y2=11 .

The Eigen vectors are and corresponding to the Eigen valuλ2x2y2es v1=10 and v211respectively.

 Step 3: Find the general solution for x→(t)

The general solution ofxt isxt=c1v1+c2tv2+v1 .

Substitute the value 10for v1 and 11for v2 in the equation xt=c1v1+c2tv2+v1as follows.


Step 3: Sketch the direction field graph ofAx→ .

Asλ1,2=0 , draw the direction field graph of the function as follows.

Hence, the formula for the solution of the systemdxdt=0100x is and the direction field graph isxt=c1+c2+c2tc2 sketched.

Recommended explanations on Math Textbooks

94% of StudySmarter users get better grades.

Sign up for free
94% of StudySmarter users get better grades.