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# clase8handouts - When does a differential equation have at...

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When does a differential equation have at least one solution? EXISTENCE. When does the model produce only one solution for a particular set of initial conditions? UNIQUENESS. M.I. Bueno Differential Equations and Linear Algebra

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Consider the differential equation y 0 = y + t . There are no solution passing through ( 0 , y ) , when y < 0, for example. M.I. Bueno Differential Equations and Linear Algebra
Consider the differential equation y 0 = t + t + - t . There are no solutions. M.I. Bueno Differential Equations and Linear Algebra

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Consider the differential equation, y 0 = y 1 / 3 There are two solutions passing through ( 0 , 0 ) . M.I. Bueno Differential Equations and Linear Algebra
Picard’s Existence and Uniqueness Theorem Given y 0 = f ( t , y ) , y ( t 0 ) = y 0 . 1 If f ( t , y ) is continuous in the rectangle R = ( a , b ) × ( c , d ) and ( t 0 , y 0 ) R , then there exists h > 0 such that the IVP has a solution in ( t 0 - h , t 0 + h ) . 2 If f y is also continuous in R , then the solution is unique. M.I. Bueno Differential Equations and Linear Algebra

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Continuity of multivariate functions Let f : D R 2 R . Let ( t 0 , y 0 ) R 2 . Then, f is continuous at ( t 0 , y 0 ) if ( t 0 , y 0 ) D , that is, f ( t 0 , y 0 ) is defined, lim ( t , y ) ( t 0 , y 0 ) f ( t , y ) exists, lim ( t , y ) ( t 0 , y 0 ) f ( t , y ) = f ( t 0 , y 0 ) . M.I. Bueno Differential Equations and Linear Algebra
M.I. Bueno Differential Equations and Linear Algebra

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Linear equations and their solutions An n th-order differential equation is linear if it can be written in the form a n ( t ) d n y dt n + a n - 1 ( t ) d n - 1 y dt n - 1 + ... + a 1 ( t ) dy dt + a 0 ( t ) y = f ( t ) , where all a i ( t ) are assumed to be defined over some common interval I . A first-order linear differential equation has the form y 0 + a ( t ) y = f ( t ) . An example is y 0 + sin ( t ) y = t + 1 . M.I. Bueno Differential Equations and Linear Algebra
A second-order linear differential equation has the form y 00 + a 1 ( t ) y 0 + a 0 ( t ) y = f ( t ) . An example is y 00 + y = t . M.I. Bueno Differential Equations and Linear Algebra

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Consider the linear differential equation given by a n ( t ) d n y dt n + a n - 1 ( t ) d n - 1 y dt n - 1 + ... + a 1 ( t ) dy dt + a 0 ( t ) y = f ( t ) .
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