clase2-handouts

clase2-handouts - Exercise A ball is dropped at rest from a...

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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 highest-order derivative that occurs. We focus on first-order 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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clase2-handouts - Exercise A ball is dropped at rest from a...

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