u 1 e u 2 e 14 This is readily solved by row reduction row reduced echelon form

# U 1 e u 2 e 14 this is readily solved by row

• 32

This preview shows page 12 - 18 out of 32 pages.

u 1 e u 2 e = - 14 0 This is readily solved by row reduction ( row reduced echelon form ) Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (12/32) Subscribe to view the full document.

Introduction Greenhouse/Rockbed Example Direction Fields and Phase Portraits Two Dimensional Model Steady State Analysis Eigenvalue Analysis Model Solution Solve Linear System 1 Solve Linear System: Write [ A : b ], so - 13 8 3 4 . . . - 14 1 4 - 1 4 . . . 0 - 8 13 R 1 -→ 4 R 2 1 - 6 13 . . . 112 13 1 - 1 . . . 0 R 2 - R 1 -→ 1 - 6 13 . . . 112 13 0 - 7 13 . . . - 112 13 - 13 7 R 2 -→ 1 - 6 13 . . . 112 13 0 1 . . . 16 R 1 + 6 13 R 2 -→ 1 0 . . . 16 0 1 . . . 16 or u e = 16 16 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (13/32) Introduction Greenhouse/Rockbed Example Direction Fields and Phase Portraits Two Dimensional Model Steady State Analysis Eigenvalue Analysis Model Solution Solve Linear System 2 Solve Linear System: Linear systems are efficiently solved in MatLab and Maple MatLab - Solving equilibrium Enter matrix, A , and vector, b Use linsolve command or inv(A)*b Augment A with b and use rref Maple - Solving equilibrium Start with(LinearAlgebra) to invoke the Linear Algebra package Enter matrix, A , and vector, b Use LinearSolve( A , b ) command or Multiply( A - 1 , b ) operation Detailed supplemental sheets are provided Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (14/32) Subscribe to view the full document.

Introduction Greenhouse/Rockbed Example Direction Fields and Phase Portraits Two Dimensional Model Steady State Analysis Eigenvalue Analysis Model Solution Solving the System of DEs 1 Model System satisfies ˙ u = Ku + b and has a steady state solution u ( t ) = u e , where Ku e = - b Make a change of variables v ( t ) = u ( t ) - u e , then ˙ v = ˙ u and ˙ v = K ( v + u e ) + b = Kv This change of variables allows considering the simpler system ˙ v = Kv Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (15/32) Introduction Greenhouse/Rockbed Example Direction Fields and Phase Portraits Two Dimensional Model Steady State Analysis Eigenvalue Analysis Model Solution Solving the System of DEs 2 Model System has a Newton’s Law of Cooling , so anticipate an exponential (decaying) solution Try a solution of the form v ( t ) = ξe λt , where ξ = [ v 1 , v 2 ] T is a constant vector, so ˙ v ( t ) = λξe λt The translated Model System ˙ v ( t ) = Kv ( t ) becomes λξe λt = K ξe λt or λξ = K ξ This is the classic eigenvalue problem ( K - λ I ) ξ = 0 , which has eigenvalues , λ , and associated eigenvectors , ξ The solution of the eigenvalue problem gives the solution of the Model System , v ( t ) = ξe λt Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (16/32) Subscribe to view the full document.

Introduction Greenhouse/Rockbed Example Direction Fields and Phase Portraits Two Dimensional Model Steady State Analysis Eigenvalue Analysis Model Solution Greenhouse Example 1 Example Model: satisfies the DE: ˙ u 1 ˙ u 2 = - 13 8 3 4 1 4 - 1 4 !  • Fall '08
• staff

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern