SystemsDifferentialEquationsForPrint.pdf

SystemsDifferentialEquationsForPrint.pdf - Systems of...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Systems of Differential Equations S. Bonnot S. Bonnot Systems of Differential Equations Examples coming from physics Goal : Physical systems where separate parts interact on each other create automatically systems of differential equations. Example : a system with 2 consecutive springs. The motion of the first mass is described by m 1 x 00 1 =- k 1 x 1 + k 2 ( x 2- x 1 ) . and the motion of the second mass at the bottom is given by m 2 x 00 2 =- k 2 ( x 2- x 1 ) . System of two equations combined together : m 1 x 00 1 =- k 1 x 1 + k 2 ( x 2- x 1 ) m 2 x 00 2 =- k 2 ( x 2- x 1 ) S. Bonnot Systems of Differential Equations Examples, continued Electric Circuits: L di 1 dt + Ri 2 = E R di 2 dt + 1 C ( i 2- i 1 = We get a system of two first order linear equations with constant coefficients . How to solve it? Eliminate one variable (say i 2 ) to obtain a single differential equation in i 1 . We know Ri 2 =- Li 1 + E , so we deduce Ri 2 =- Li 00 1 + E . Replace i 2 and i 2 in the second equation, and obtain (- Li 00 1 + E ) + 1 RC (- Li 1 + E )- 1 C = 0. Solve this equation to get i 1 and then use Ri 2 =- Li 1 + E to obtain i 2 . S. Bonnot Systems of Differential Equations General case: system of first order equ. with constant coefficients Given the system a 1 x + b 1 y + c 1 x + d 1 y = f 1 ( t ) a 2 x + b 2 y + c 2 x + d 2 y = f 2 ( t ) let’s try to eliminate y by taking b 2 . ( Line 1 )- b 1 ( Line 2 ) . we obtain ( a 1 b 2- a 2 b 1 ) x + ( c 1 b 2- c 2 b 1 ) x + ( d 1 b 2- d 2 b 1 ) y = b 2 f 1 ( t )- b 1 f 2 ( t ) this equation is of type Ax + Bx + Cy = F ( t ) . Case C = 0. Solve Ax + Bx = F ( t ) then substitute in either equation to get y . Case C 6 = 0. We know y = (- Ax- Bx + F ( t )) / C , so we know y and we can substitute y , y in Equ.1 or Equ.2. S. Bonnot Systems of Differential Equations Solving systems of first order equ. with constant coefficients Example : 3 x + 2 y + x + y = e t 6 x + 4 y + x = t We eliminate y by taking Line 2- 2 Line 1, to obtain- x- 2 y = t- 2 e t . Observe that by accident we also eliminated x . So far we know y = 1 2 (- x- t + 2 e t ) , so we also deduce y = 1 2 (- x- 1 + 2 e t ) . Substitute y , y in Equ.1 to get a first order equ. in x alone: 3 x + (- x- 1 + 2 e t ) + x + ( 1 2 (- x- t + 2 e t )) = e t ....
View Full Document

This document was uploaded on 04/01/2011.

Page1 / 19

SystemsDifferentialEquationsForPrint.pdf - Systems of...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online