week_04_4.1 - MATH2860U Chapter 4 1 HIGHER-ORDER...

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MATH2860U: Chapter 4 1 HIGHER-ORDER DIFFERENTIAL EQUATIONS Linear Differential Equations: Basic Theory (4.1, pg. 118) Recall: So far, we’ve only discussed first-order equations. Let us now move on to solving higher-order differential equations, as we have already seen that these arise frequently in applications. Definition: An n th order linear differential equation is an equation of the form ) ( ) ( ) ( ) ( ) ( 0 1 1 1 1 x g y x a dx dy x a dx y d x a dx y d x a n n n n n n If we want to solve it subject to initial conditions 1 0 1 1 0 0 0 ) ( , , ) ( , ) ( n n y x y y x y y x y then this is an n th order initial-value problem . Before solving such equations, let’s take some time to develop the theory first. Theorem: Let ) ( ), ( , ), ( ), ( 0 1 1 x a x a x a x a n n and ) ( x g be continuous on an interval I , and let 0 ) ( x a n for every x in this interval. If 0 x x is any point in this interval, then a solution ) ( x y of the initial value problem ) ( ) ( ) ( ) ( ) ( 0 1 1 1 1 x g y x a dx dy x a dx y d x a dx y d x a n n n n n n subject to 1 0 1 1 0 0 0 ) ( , , ) ( , ) ( n n y x y y x y y x y exists and is unique. Example: What is the largest interval on which 0 ) 5 ( ) 3 ( 2 y t y y t t subject to 6 ) 1 ( , 0 ) 1 ( y y is guaranteed to have a unique solution?
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MATH2860U: Chapter 4 2 Example: What is the largest interval on which x y y x y x y x tan 2 2 2 3
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week_04_4.1 - MATH2860U Chapter 4 1 HIGHER-ORDER...

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