lec14h - Math 16b Thomas Scanlon Autumn 2010 Thomas Scanlon Math 16b Introduction to ODEs An(ordinary dierential equation is an equation involving a

lec14h - Math 16b Thomas Scanlon Autumn 2010 Thomas Scanlon...

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Unformatted text preview: Math 16b Thomas Scanlon Autumn 2010 Thomas Scanlon Math 16b Introduction to ODEs An (ordinary) di erential equation is an equation involving a function and its derivatives. That is, for functions P ( x , x 1 ,..., x n ) and Q ( x ,..., x n ) of n + 1 variables, we say that the function f ( t ) (of one variable) satis es the di erential equation P ( y , y ,..., y ( n ) ) = Q ( f ( t ) ,..., f ( n ) ( t )) if P ( f ( t ) , f ( t ) ,..., f ( n ) ( t )) ≡ Thomas Scanlon Math 16b Examples The function f ( t ) = e t satis es the di erential equation y = y . The constant function g ( t ) ≡ 5 satis es the di erential equation y = 0. The functions h ( t ) = sin ( t ) and k ( t ) = cos ( t ) satisfy the di erential equation y 00 + y = 0. The function ‘ ( t ) = ln ( t ) satis es- ( y ) 2 = y 00 . Since ‘ ( t ) = 1 / t and ‘ 00 ( t ) =- 1 / t 2 , so that ‘ 00 =- 1 / t 2 =- ( 1 / t ) 2 =- ( ‘ ) 2 Thomas Scanlon Math 16b Initial value problems An initial value problem is a di erential equation given together with some requirements on the value of the function (or possibly some of its derivatives) at certain points....
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