The Computable Differential Equation Lecture ... - Bruce E. Shapiro

168 CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS
so the original DAE is equivalent to the ODE
u ′ = v (9.10)
v ′ = w (9.11)
w ′ = − v(ew + we u )
ue w + e u (9.12)
which gives explicit expressions for each of the derivatives in terms of the variables.
This type of DAE is said to be of index 1 because a single differentiation was sufficient
to convert the equation to a system of ODEs.
Definition 9.1. The index of a DAE is the minimum number of times all or part of
the DAE must be differentiated with respect to t to be able to determine y ′ explicitly
as a function of t and y.
Example 9.1. Show that the equations for the unit pendulum
y ′′ = g − T cos θ (9.13)
x ′′ = −T sin θ (9.14)
1 = x 2 + y 2 (9.15)
where T is the unknown tension in the pendulum rope, form an index-3 system.
Solution. We can rewrite this as a first-order system by defining cartesian velocity
components u = x ′ and v = y ′ , and by observing that because of the constraint
Hence the first order DAE becomes
y = cos θ (9.16)
x = sin θ (9.17)
x ′ = u (9.18)
y ′ = v (9.19)
u ′ = −T x (9.20)
v ′ = g − T y (9.21)
0 = x 2 + y 2 − 1 (9.22)
where T is an unknown variable. To get a differential equation for T we must
differentiate the constraint three times, as follows. First, we obtain
Differentiating a second time,
0 = xx ′ + yy ′ (9.23)
0 = xu + yv (9.24)
0 = xu ′ + x ′ u + yv ′ + y ′ v (9.25)
0 = −x 2 T + u 2 + yg − T y 2 + v 2 (9.26)
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007