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**Unformatted text preview: **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 Consider the differential equation y = y + t . There are no solution passing through ( , y ) , when y < 0, for example. M.I. Bueno Differential Equations and Linear Algebra Consider the differential equation y = t + t + - t . There are no solutions. M.I. Bueno Differential Equations and Linear Algebra Consider the differential equation, y = y 1 / 3 There are two solutions passing through ( , ) . M.I. Bueno Differential Equations and Linear Algebra Picards Existence and Uniqueness Theorem Given y = f ( t , y ) , y ( t ) = y . 1 If f ( t , y ) is continuous in the rectangle R = ( a , b ) ( c , d ) and ( t , y ) R , then there exists h > 0 such that the IVP has a solution in ( t- h , t + h ) . 2 If f y is also continuous in R , then the solution is unique. M.I. Bueno Differential Equations and Linear Algebra Continuity of multivariate functions Let f : D R 2 R . Let ( t , y ) R 2 . Then, f is continuous at ( t , y ) if ( t , y ) D , that is, f ( t , y ) is defined, lim ( t , y ) ( t , y ) f ( t , y ) exists, lim ( t , y ) ( t , y ) f ( t , y ) = f ( t , y ) . M.I. Bueno Differential Equations and Linear Algebra M.I. Bueno Differential Equations and Linear Algebra 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 ( 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 + a ( t ) y = f ( t ) . An example is y + 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 + a ( t ) y = f ( t ) ....

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