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

176 CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS
Example 9.3. Show that (y ′ ) 2 = t 2 , y(0) = 0, is a Hessenberg form of size 1 and
that its solution is not unique.
Solution. To write the equation in Hessenberg form we add the new variable x = y ′ .
Then our system reads as
y ′ = f(t, y, x) = x (9.95)
0 = g(t, y, x) = x 2 − t 2 (9.96)
Since ∂g/∂x = 2x is not identically zero the Jacobian is nonsingular. To find the
solutions we observe that any solution to either of the following equations satisfies
the original equaiton:
y ′ = t (9.97)
y ′ = −t (9.98)
Hence both y = t 2 /2 and y = −t 2 /2 are solutions. Hence
( x
=
y)
( ) t
t 2 /2
or
( ) −t
−t 2 /2
(9.99)
The Hessenberg form of size 2 can be written as
y ′ = f(t, y, x) (9.100)
0 = g(t, y) (9.101)
where the product of Jacobians
is nonsingular
The Hessenberg form of size 3 is
∂g ∂f
∂y ∂x
(9.102)
x ′ = f(t, x, y, z) (9.103)
y ′ = g(t, x, y) (9.104)
0 = h(t, y) (9.105)
where the product of Jacobians
is nonsingular
∂h ∂g ∂f
∂y ∂x ∂z
(9.106)
Example 9.4. Show that pendulum equations discussed in example 9.1 is an Hessenberg
Index-3 DAE.
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007