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

68 CHAPTER 3. APPROXIMATE SOLUTIONS
By the ODE (y ′ = f(t, y)) and the mean value theorem (in the second argument)
d
[y(t) − ỹ(t)] = f(t, y(t)) − f(t, ỹ(t)) (3.217)
dt
for some η(t) between y(t) and ỹ(t). Hence
= [y(t) − ỹ(t)] ∂ f(t, η(t)) (3.218)
∂y
∂p(t, y 0 )
∂t
=
y(t) − ỹ(t)
y 0 − ỹ 0
∂
f(t, η(t)) (3.219)
∂y
= p(t, y 0 ) ∂ f(t, η(t)) (3.220)
∂y
where η(t) → y(t) as ỹ(t) → y(t). Since p ≠ 0 so long as y 0 ≠ ỹ 0 , we may divide by
p:
1 ∂p(t, y 0 )
= ∂ f(t, η(t)) (3.221)
p(t , y 0 ) ∂t ∂y
ln p(t , y 0 ) =
∫ t
t 0
∂
f(s, η(s))ds (3.222)
∂y
The constant of integration is determined from p(t 0 , y 0 ) = 1, and
{∫ t
}
∂
p(t , y 0 ) = exp
t 0
∂y f(s, η(s))ds
(3.223)
By the definition of p,
∂y(t, ỹ 0 )
∂y 0
= lim p(t, y 0 ) (3.224)
y 0 →ỹ 0
{∫ t
}
∂
= exp
∂y f(s, ỹ(s))ds (3.225)
t 0
which is continuous in both arguments.
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007