Systems of Ordinary Differential Equations MATH 365 Ordinary Differential Equations J Robert Buchanan Department of Mathematics Fall 2018
Objectives Many physical problems involve a number of separate elements linked together in some fashion Mathematical models of these situations consist of a system of two or more differential equations In this lesson we will: explore several examples of systems of ordinary differential equations, and learn to write systems of ordinary differential equations as first-order equations
Two Degree of Freedom Spring-Mass System Consider the spring-mass system depicted below For the moment ignore the damping (ie, let c 1 = c 2 = c 3 = 0) and derive a system of ODEs to describe the motion of the spring-mass system
Solution m 1 d 2 x 1 dt 2 = k 2 (x 2 x 1 ) k 1 x 1 + F 1 (t) = (k 1 + k 2 )x 1 + k 2 x 2 + F 1 (t) m 2 d 2 x 2 dt 2 = k 3 x 2 k 2 (x 2 x 1 ) + F 2 (t) = k 2 x 1 (k 1 + k 3 )x 2 + F 2 (t)
Parallel LRC Circuit Consider the parallel LRC circuit shown below where V is the voltage and I is the current A system of ODEs describing this circuit is di dt dv dt = V L = I C V R C
Predator-Prey Model Consider an ecosystem in which a predator species P and a prey species H live A mathematical model for their interaction could be dh dt dp dt = a 1 H b 1 H P = a 2 P + b 2 H P where a 1, a 2, b 1, and b 2 are positive constants
Systems of First-Order ODEs Definition A system of simultaneous first-order ODEs has the general form x 1 = F 1 (t, x 1, x 2,, x n ) x 2 = F 2 (t, x 1, x 2,, x n ) x n = F n (t, x 1, x 2,, x n ) where each x k is a function of t If each of the F k is a linear function of x 1, x 2,, x n, then the system of equations is said to be linear, otherwise it is nonlinear
Observation Ordinary differential equations of higher order can always be transformed into a system of first-order ODEs Example Consider the second-order linear ODE: m u (t) + γ u (t) + k u(t) = F(t) Let x 1 = u(t) and x 2 = u (t) and re-write the second-order ODE as a system of two first-order ODEs
Solution Second-order linear ODE: System of first-order ODEs: m u (t) + γ u (t) + k u(t) = F(t) x 1 = u = x 2 x 2 = u = 1 m (F(t) k u γ u ) = 1 m (F(t) k x 1 γ x 2 )
General Case An arbitrary nth-order ordinary differential equation: y (n) = F(t, y, y, y,, y (n 1) ) can be transformed into a system of n first-order equations by defining x 1 = y, x 2 = y, x 3 = y,, x n = y (n 1) x 1 = x 2 x 2 = x 3 x n 1 = x n x n = F(t, x 1, x 2,, x n )
Systems of First-order ODEs A system of simultaneous first order ordinary differential equations has the general form: x 1 = F 1 (t, x 1, x 2,, x n ) x 2 = F 2 (t, x 1, x 2,, x n ) x n = F n (t, x 1, x 2,, x n ) Definition A solution to the system on the interval α < t < β is a set of functions x 1 = φ 1 (t), x 2 = φ 2 (t),, x n = φ n (t) that are differentiable for α < t < β and satisfy the system of ODEs
Initial Value Problems x 1 = F 1 (t, x 1, x 2,, x n ) x 2 = F 2 (t, x 1, x 2,, x n ) x n = F n (t, x 1, x 2,, x n ) x 1 (t 0 ) = x1 0 x 2 (t 0 ) = x2 0 If, in addition to the given system of differential equations there are also initial conditions, this system forms an initial value problem x n (t 0 ) = x 0 n
Example Consider the initial value problem y + y = 0 y(0) = 0 y (0) = 1 Write this initial value problem as a system of equations and verify that solves the system x 1 (t) = sin t x 2 (t) = cos t
Existence and Uniqueness of Solutions x 1 = F 1 (t, x 1, x 2,, x n ) x 2 = F 2 (t, x 1, x 2,, x n ) x n = F n (t, x 1, x 2,, x n ) x 1 (t 0 ) = x1 0 x 2 (t 0 ) = x2 0 x n (t 0 ) = xn 0 Theorem Suppose F 1,, F n are F 1 / x 1,, F 1 / x n,, F n / x n are continuous in the region R of (t, x 1, x 2,, x n )-space defined by α < t < β, α 1 < x 1 < β 1,, α n < x n < β n and let the point (t 0, x 0 1, x 0 2,, x 0 n ) be contained in R Then in some interval (t 0 h, t 0 + h) there exists a unique solution x 1 = φ 1 (t), x 2 = φ 2 (t), x n = φ n (t) to the initial value problem
Linear Systems If each of the functions F 1,, F n is a linear function of the dependent variables x 1,, x n, then the system of ordinary differential equations is said to be linear; otherwise, it is nonlinear x 1 (t) = p 11(t)x 1 (t) + + p 1n (t)x n (t) + g 1 (t) x 2 (t) = p 21(t)x 1 (t) + + p 2n (t)x n (t) + g 2 (t) x n(t) = p n1 (t)x 1 (t) + + p nn (t)x n (t) + g n (t) If each of the functions g 1 (t),, g n (t) is zero, the system is said to be homogeneous; otherwise it is non-homogeneous
Another Existence/Uniqueness Result x 1(t) = p 11 (t)x 1 (t) + + p 1n (t)x n (t) + g 1 (t) x 2(t) = p 21 (t)x 1 (t) + + p 2n (t)x n (t) + g 2 (t) x n(t) = p n1 (t)x 1 (t) + + p nn (t)x n (t) + g n (t) x 1 (t 0 ) = x 0 1 x 2 (t 0 ) = x 0 2 x n (t 0 ) = x 0 n Theorem If the functions p 11, p 12,, p nn, g 1,, g n are continuous on an open interval α < t < β, then there exists a unique solution x 1 = φ 1 (t),, x n = φ n (t) to the linear system where t 0 (α, β) and x 0 1,, x 0 n are any prescribed numbers The solution exists for α < t < β
Homework Read Section 71 Exercises: 1, 3, 5, 15, 16