2130_Chapter7_Notes

pnn t g1 t gn t are continuous on an open

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Unformatted text preview: = x0 , x2 (t0 ) = x0 , . . . , xn (t0 ) = x0 . n 1 2 Questions When does a solution exist? If a solution exists, is it unique? Theorem 7.1.2 If the coefficient functions p11 (t), p12 (t), . . . , pnn (t), g1 (t), . . . , gn (t) are continuous on an open interval I = (α, β ), then the system with initial conditions has a unique solution x1 = φ1 (t), x2 = φ2 (t), . . . , xn = φn (t) throughout the entire interval (α, β ). 5/9 3 Example 2 x′ = 3x1 − 2x2 , x1 (0) = 3 1 x′ = 2x1 − 2x2 , x2 (0) = 1/2 2 Solve this system by converting to a second order linear equation: x′ = 3x1 − 2x2 =⇒ x2 = 1 (3x1 − x′ ) =⇒ x′ = 1 (3x′ − x′′ ). 1 1 2 1 1 2 2 Now substitute for x2 and x′ in the second equation: 2 x′ = 2x1 − 2x2 =⇒ 2 1 ′ 2 (3x1 − x′′ ) = 2x1 − (3x1 − x′ ) 1 1 =⇒ 3x′ − x′′ = 4x1 − 6x1 + 2x′ 1 1 1 =⇒ x′′ − x′ − 2x1 = 0. 1 1 Note: the original system was homogeneous, and so the resulting equation is also homogeneous. 6/9 Exa 2 (slid...
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