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Unformatted text preview: MA 36600 FINAL REVIEW Chapter 7
§7.1: Introduction.
• A system of ﬁrst order equations is a collection of initial value problems dxk
= Gk (t, x1 , x2 , . . . , xn ) ,
dt
k = 1, 2, . . . , m. xk (t0 ) = x0 ;
k We say that such a system is a system of ﬁrst order linear equations if we can write
Gk (t, x1 , x2 , . . . , xn )
k = 1, 2, . . . , m.
= pk1 (t) x1 + pk2 (t) x2 + · · · + pkn (t) xn + gk (t); If at least one of the Gk (t, x1 , x2 , . . . , xn ) is not in the form above, we call the system nonlinear. We
call a linear system homogeneous if each gk (t) is the zero function. Otherwise, we call the system
nonhomogeneous.
• We can always express an nth order linear equation as a system of ﬁrst order equations: In general,
an nth order equation is an ordinary diﬀerential equation in the form
y (n) = G t, y, y (1) , . . . , y (n−1)
where the notation y (k) denotes the k th derivative; and we have the initial conditions
y (t0 ) = y0 ,
If we make the substitutions x1 = y (1) x2 = y =⇒
.
. . xn = y (n−1) (1) y (1) (t0 ) = y0 , ... (n−1) y (n−1) (t0 ) = y0 x0 = y0
1 x = x2
1
.
.
. (1) x n−1 = xn x = G (t, x1 , . . . , xn )
n • Assume that there are as many equations
R = (t, x1 , . . . , xn ) ∈ Rn+1 . where x0 = y0
2
.
.
. (n−1) x0 = y0
n as variables i.e., that m = n. Say that there is a region
α<t<β
αk < xk < βk for k = 1, 2, . . . , n such that
i. Each Gk (t, x1 , x2 , . . . , xn ) is continuous on R. (There are m functions to consider here.)
∂ Gi
is continuous on R. (There are m · n functions to consider.)
ii. Each
∂ xj
iii. t0 , x0 , . . . , x0 ∈ R.
n
1
Then there exists a subregion R ⊆ R such that the system of diﬀerential equations has a unique
solution. This is known as the Existence and Uniqueness Theorem for Nonlinear Systems.
• Consider a system of ﬁrst order linear equations dxk = pk1 (t) x1 + pk2 (t) x2 + · · · + pkn (t) xn + gk (t),
dt
k = 1, 2, . . . , m. xk (t0 ) = x0 ;
k
Again, for simplicity assume that m = n. Say that there is an interval
α<t<β
I= t∈R
1 ...
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 Spring '09
 EdrayGoins
 Equations

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