hwsoln1spring08

hwsoln1spring08 - Differential Equations/Linear Algebra -...

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Unformatted text preview: Differential Equations/Linear Algebra - MTH 2201/2202 Homework 1 Name: 01/15/2008 1. Given y ( t ) = e- t 2 . Then y ( t ) =- 1 2 e- t 2 . Substituting y ( t ) and y ( t ) in the given differential equation, we obtain 2 y + y = 2(- 1 2 ) e- t 2 + e- t 2 = 0 . 2. Consider x 2 y + y 2 = 1. Differentiating on both sides with respect to y , we get 2 xy dx dy + x 2 + 2 y = 0 2 xydx + ( x 2 + 2 y ) dy = 0 . 3. Given y ( t ) = e- t 2 R t e u 2 du + c 1 e- t 2 . Then, y ( t ) = e- t 2 e t 2 +(- 2 t ) e- t 2 Z t e u 2 du + c 1 e- t 2 (- 2 t ) = 1- 2 te- t 2 Z t e u 2- c 1 e- t 2 (2 t ) Substituting y ( t ) and y ( t ) in the given differential equation, we obtain y ( t ) + 2 ty = 1. 4. Given x ( t ) = e- 2 t + 3 e 6 t and y ( t ) =- e- 2 t + 5 e 6 t . Then, x ( t ) =- 2 e- 2 t + 18 e 6 t and y ( t ) = 2 e- 2 t + 30 e 6 t . Substituting x ( t ), 1 y ( t ), x ( t ) and y ( t ), we get dx dt = x + 3 y, dy dt = 5 x + 3 y....
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hwsoln1spring08 - Differential Equations/Linear Algebra -...

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