University:
Georgia Southern University
Course:
MATH 3230 | Ordinary Differential Equations
Academic year:

2024
Views:

158

Pages:

5
Author:

Jalen W.

Definition Let t be a real variable, and suppose we have the following. A system of n linear ODEs has the following form... y10 = a1,1 y1 + a1,2 y2 + · + a1,n yn + b1 y20 = a2,1 y1 + a2,2 y2 + · + a2,n yn + b2 .. . yn0 = an,1 y1 + an,2 y2 + · + an,n yn + bn Definition ...where I ai,j (t) are real-valued functions of t. I b1 (t), b2 (t), . . . , bn (t) are real-valued functions of t I y1 (t), y2 (t), . . . , yn (t) are real-valued differentiable functions System of 2 linear ODEs Just have x(t) and y (t). x 0 = a1,1 x + a1,2 y + b1 y 0 = a2,1 x + a2,2 y + b2 Connection with Higher-order Linear ODEs Suppose we have the second order-linear ODE a(t)y 00 + b(t)y 0 + c(t)y = f (t). Let x = y 0 so that x 0 = y 00 = − b(t) a(t) x − have the following linear system x0 y0 = = − b(t) a(t) x x − c(t) a(t) y c(t) a(t) y + + f (t) a(t) . f (t) a(t) Then, we Example Convert the second order linear ODE to a system of 2 linear ODEs. 2y 00 + 3y 0 + y = 12t

Section 9.1: Relationship between Higher-order Linear ODEs

Section 9.1: Relationship between Higher-order Linear ODEs - Page 3

Section 9.1: Relationship between Higher-order Linear ODEs - Page 4

Section 9.1: Relationship between Higher-order Linear ODEs - Page 5

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