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Unformatted text preview: n open interval I = (α, β ) containing t0 , then
the initial value problem
x′ = P(t)x + g(t), x(t0 ) = x0
has a unique solution x = φ(t) throughout the entire interval (α, β ).
Note: The functions P(t) and g(t) are said to be continuous on (α, β ) when each of the individual scalar
p11 (t), . . . , pnn (t), g1 (t), . . . , gn (t)
is continuous there.
3 / ?? 2 Homogeneous System
The system is homogeneous if g(t) = 0 =⇒ x′ = P(t)x.
If x(1) , . . . , x(n) are solutions, then c1 x(1) + · · · + cn x(n) is also a solution.
The vectors x(1) , . . . , x(n) are independent at t0 ⇐⇒ the matrix (1) (n) x1 (t0 ) ··· x1 (t0 ) X = x(1) (t0 ) · · · x(n) (t0 ) = .
. .. . (1) .
. (n) xn (t0 ) ··· xn (t0 ) is invertible ⇐⇒ det X = 0. This determinant is known as the Wronskian W x(1) , . . . , x(n) (t0 ) = W (t0 ).
If x(1) , . . . , x(n) are linearly independent solutions, then the general solution is
x = c1 x(1) + · · · + cn x(n) .
4 / ?? Summary
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This note was uploaded on 03/19/2014 for the course APMA 2130 taught by Professor Bernardfulgham during the Fall '09 term at UVA.
- Fall '09