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Unformatted text preview: Exercise A ball is dropped at rest from a certain height. Its velocity after t seconds due to the earth’s gravitational field is given by v ( t ) = 32 . 2 t ft/sec , (when air resistance is ignored). How far has the ball fallen after t seconds? M.I. Bueno Differential Equations and Linear Algebra A differential equation is an equation containing derivatives. The order of the equation refers to the highestorder derivative that occurs. We focus on firstorder differential equations that can be written as dy dt = f ( t , y ) . If t denotes time, then we have a dynamical system . The variable y stands for an unknown function. M.I. Bueno Differential Equations and Linear Algebra Solution of a DE Let dy dt = . 03 y . y ( t ) = . 9 e . 03 t is a solution because dy ( t ) dt = d ( . 9 e . 03 t ) dt = . 9 · . 03 · e . 03 t = . 03 y ( t ) , for all t ∈ R . M.I. Bueno Differential Equations and Linear Algebra A solution is a function that must satisfy the DE for all values of t , that is, y ( t ) is a solution of the equation dy dt = f ( t , y ) if substituting y by y ( t ) the equation becomes an identity on an appropriate domain for t . Example: y ( t ) = 5 e 3 t is a solution of dy dt = 3 y . d ( 5 e 3 t ) dt = 5 · 3 · e 3 t . 5 · 3 · e 3 t = 3 y ( t ) . This is true for all t . Thus, y ( t ) is a solution in (∞ , ∞ ) ....
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 Fall '08
 JACOBS
 Linear Algebra, Derivative, 2 m, 0 0 0 1 2 M, 0.2 0 0.2 0.4 0.6 0.8 1 M

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