1550078481-Ordinary_Differential_Equations__Roberts_

314 Ordinary Differential Equations

In chapter 3, we derived a syst em of two, first-order differential equations
for t he quantities, q 1 (t) and q 2 (t), of dye in two tanks as a function of time t.
The system was

dq1 = .5 - q1

(2) dt^30

dq2 -q1 --q2

dt 30 15

In t hi s system q 1 and q 2 are the dependent variables and t is the independent
variable. Recall we were able to solve this system by solving t he first equation
explicit ly for q 1 , substituting this result into the second equation and then
solving the resulting differential equation for q2.

In chapter 6 we solved two coupled spring-mass systems, a double pendulum
system, a system for the path of an electron, and systems resulting from
compartmental analysis by rewriting each system as a single higher order
differential equation. The first coupled spring-mass system was the following
system of two, second-order differential equations

m1y~ = - k1Y1 + k2(Y2 - Y1)

m2y~ = - k2(Y2 - Y1).

(3)

Here m 1 and m 2 are the masses attached to the springs with spring constants
k1 and k2 respectively and y 1 and Y2 are the displacements of the masses from
equilibrium. (See Figure 6.7.) We can rewrite this system as a system offirst-

order differential equations in the following manner. Let u 1 = y 1 , u2 = y~,

u3 = y2, and u 4 = y~. So u 1 is the position of the first mass and u2 is its

velocity. While u 3 is the position of the second mass and u 4 is its velocity.
Differentiating u 1 = y 1 , u2 = y~, u3 = Y2, and u4 = y~ and then substituting
for Y1, y~, y2, and y~ in terms of u1, u2, u3, and U4, we find

U~ = y~ = U2

I II -k1Y1 + k2(Y2 - Y1)

U2 = Y1 =

m1

U~ = y~ = U4

I II -k2(Y2 - Y1)

U4 = Y2 =

m2

Thus, the system of two, second-order differential equations (3) is equivalent
to the following system of four, first-order differential equations

U1 I = U2

(4)

, -k1u 1 + k2(u3 - u1)

U2 =

m1

I

U3 = U4

, -k2(u3 - u1)

U4 =