2302show - Diff Eqs Supplement J Mart´ ınez University of...

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Unformatted text preview: Diff. Eqs., Supplement J. Mart´ ınez University of Florida Spring 2010 Non-homogeneous Equations Laplace Transforms Wronskian Undetermined Coefficients Exercise 1.1 (p. 216; 35) Consider y 00 + p ( t ) y + q ( t ) y = g ( t ) , with solutions 1 + t , 1 + 2 t , and 1 + 3 t 2 . Then y 1 ( t ) = t and y 2 ( t ) = 3 t 2- t are independent solutiond to the corresponding homogeneous equation. W = W [ y 1 , y 2 ] = ( 6 t- 1 ) t- ( 3 t 2- t ) = 3 t 2 , and W = 6 t . By Abel’s Formula, p ( t ) =- W W =- 2 t . Plugging the first solution into the homogeneous equation, and then the original, yields q ( t )- 2 t 2 , g ( t ) =- 2 t 2 . The solution is y ( t ) = a 1 t + a 2 ( 3 t 2- t ) + t + 1 . J. Mart´ ınez MAP 2302 Non-homogeneous Equations Laplace Transforms Wronskian Undetermined Coefficients Exercise 1.1 (p. 216; 35) Consider y 00 + p ( t ) y + q ( t ) y = g ( t ) , with solutions 1 + t , 1 + 2 t , and 1 + 3 t 2 . Then y 1 ( t ) = t and y 2 ( t ) = 3 t 2- t are independent solutiond to the corresponding homogeneous equation. W = W [ y 1 , y 2 ] = ( 6 t- 1 ) t- ( 3 t 2- t ) = 3 t 2 , and W = 6 t . By Abel’s Formula, p ( t ) =- W W =- 2 t . Plugging the first solution into the homogeneous equation, and then the original, yields q ( t )- 2 t 2 , g ( t ) =- 2 t 2 . The solution is y ( t ) = a 1 t + a 2 ( 3 t 2- t ) + t + 1 . J. Mart´ ınez MAP 2302 Non-homogeneous Equations Laplace Transforms Wronskian Undetermined Coefficients Exercise 1.1 (p. 216; 35) Consider y 00 + p ( t ) y + q ( t ) y = g ( t ) , with solutions 1 + t , 1 + 2 t , and 1 + 3 t 2 . Then y 1 ( t ) = t and y 2 ( t ) = 3 t 2- t are independent solutiond to the corresponding homogeneous equation. W = W [ y 1 , y 2 ] = ( 6 t- 1 ) t- ( 3 t 2- t ) = 3 t 2 , and W = 6 t . By Abel’s Formula, p ( t ) =- W W =- 2 t . Plugging the first solution into the homogeneous equation, and then the original, yields q ( t )- 2 t 2 , g ( t ) =- 2 t 2 . The solution is y ( t ) = a 1 t + a 2 ( 3 t 2- t ) + t + 1 . J. Mart´ ınez MAP 2302 Non-homogeneous Equations Laplace Transforms Wronskian Undetermined Coefficients Cramer’s Rule Applied to the “target” solution y = v 1 y 1 + v 2 y 2 + ··· + v n y n , one uses the system v 1 y 1 + ··· + v n y n = v 1 y 1 + ··· + v n y n = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . = v 1 y ( n- 2 ) 1 + ··· + v n y ( n- 2 ) n = v 1 y ( n- 1 ) 1 + ··· + v n y ( n- 1 ) n = g J. Mart´ ınez MAP 2302 Non-homogeneous Equations Laplace Transforms Wronskian Undetermined Coefficients More Cramer’s Rule The Wronskian W = W [ y 1 ,..., y ] is the determinant of an n × n matrix y 1 y 2 ... y n y 1 y 2 ... y n ... ... ... ......
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2302show - Diff Eqs Supplement J Mart´ ınez University of...

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