This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 334 Lecture #29 7.1,7.2,7.3: Introduction to Systems of First Order ODEs Example. (Converting a second order ODE into a system of two first order ODEs.) An IVP for a massspring system is mu 00 + u + ku = F ( t ) u (0) = u , u (0) = u . Define two new dependent variables x 1 = u and x 2 = u . The derivatives of these new dependent variables with respect to the independent variable t are x 1 = u = x 2 , x 2 = u 00 = F ( t ) u ku m = F ( t ) x 2 kx 1 m . [The original second order ODE in u was solved for u 00 and substituted into the second first order ODE.] In matrix form, this system of first order ODEs is x 1 x 2 = 1 k/m /m x 1 x 2 + F ( t ) /m . This conversion of a second order ODE into a system of first order ODEs illustrates how to convert of an n th order ODE into a system of n first order ODEs. Example. (Obtaining systems of first order ODEs through modeling.) A dissolved chemical is mixed in a coupled system of two tanks, each of which contains 100 gallons of water....
View
Full
Document
 Spring '11
 Smith
 Math

Click to edit the document details