Lecture Notes in Differential Equations - Bruce E. Shapiro

226 LESSON 26. GENERAL EXISTENCE THEORY*
to a system.
We create a system by defining the variables
Then the differential equation becomes
which we can rewrite as
We then define functions f, and g,
so that our system can be written as
x 1 = y
x 2 = y ′ (26.62)
x ′ 2 + 4t 3 x 2 + x 3 1 = sin t (26.63)
x ′ 2 = sin t − x 3 1 − 4t 3 x 2 (26.64)
f(x 1 , x 2 ) = sin t − x 3 1 − 4t 3 x 2
g(x 1 , x 2 ) = x 2
(26.65)
x ′ 1 = f(x 1 , x 2 )
x ′ 2 = g(x 1 , x 2 )
(26.66)
with initial condition
x 1 (0) = 1
x 2 (0) = 1
(26.67)
It is common to define a vector x = (x 1 , x 2 ) and a vector function
F(x) = (f(x 1 , x 2 ), g(x 1 , x 2 )) (26.68)
Then we have a vector initial value problem
}
x ′ (t) = F(x) = (sin t − x 3 1 − 4t 3 x 2 , x 2 )
x(0) = (1, 1)
(26.69)
Since the set of all differentiable functions on R 2 is a vector space, our
theorem on vector spaces applies. Even though we proved theorem (26.50)
for first order equations every step in the proof still works when y and f
become vectors. On any closed rectangle surrounding the initial condition
F and ∂F/∂x i is bounded, continuous, and differentiable. So there is a
unique solution to this initial value problem.