final_review (dragged)

final_review (dragged) - MA 36600 FINAL REVIEW Chapter 7...

Info iconThis preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: MA 36600 FINAL REVIEW Chapter 7 §7.1: Introduction. • A system of first order equations is a collection of initial value problems dxk = Gk (t, x1 , x2 , . . . , xn ) , dt k = 1, 2, . . . , m. xk (t0 ) = x0 ; k We say that such a system is a system of first order linear equations if we can write Gk (t, x1 , x2 , . . . , xn ) k = 1, 2, . . . , m. = pk1 (t) x1 + pk2 (t) x2 + · · · + pkn (t) xn + gk (t); If at least one of the Gk (t, x1 , x2 , . . . , xn ) is not in the form above, we call the system nonlinear. We call a linear system homogeneous if each gk (t) is the zero function. Otherwise, we call the system nonhomogeneous. • We can always express an nth order linear equation as a system of first order equations: In general, an nth order equation is an ordinary differential equation in the form ￿ ￿ y (n) = G t, y, y (1) , . . . , y (n−1) where the notation y (k) denotes the k th derivative; and we have the initial conditions y (t0 ) = y0 , If we make the substitutions x1 = y (1) x2 = y =⇒ . . . xn = y (n−1) (1) y (1) (t0 ) = y0 , ... (n−1) y (n−1) (t0 ) = y0 x0 = y0 1 x￿ = x2 1 . . . (1) x￿ n−1 = xn x￿ = G (t, x1 , . . . , xn ) n • Assume that there are as many equations ￿ R = (t, x1 , . . . , xn ) ∈ Rn+1 . where x0 = y0 2 . . . (n−1) x0 = y0 n as variables i.e., that m = n. Say that there is a region ￿ ￿ ￿ α<t<β ￿ ￿ αk < xk < βk for k = 1, 2, . . . , n such that i. Each Gk (t, x1 , x2 , . . . , xn ) is continuous on R. (There are m functions to consider here.) ∂ Gi is continuous on R. (There are m · n functions to consider.) ii. Each ∂ xj ￿ ￿ iii. t0 , x0 , . . . , x0 ∈ R. n 1 Then there exists a subregion R￿ ⊆ R such that the system of differential equations has a unique solution. This is known as the Existence and Uniqueness Theorem for Nonlinear Systems. • Consider a system of first order linear equations dxk = pk1 (t) x1 + pk2 (t) x2 + · · · + pkn (t) xn + gk (t), dt k = 1, 2, . . . , m. xk (t0 ) = x0 ; k Again, for simplicity assume that m = n. Say that there is an interval ￿ ￿ ￿ ￿ ￿α<t<β I= t∈R￿ 1 ...
View Full Document

Ask a homework question - tutors are online