lecture_19 (dragged) 2

lecture_19 (dragged) 2 - y ( t ) = c 1 e t + c 2 e 4 t + 2...

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MA 36600 LECTURE NOTES: MONDAY, MARCH 2 3 for some constants c 1 and c 2 . Variation of Parameters Recap. Recall that we wish to Fnd the general solution to the linear second order di±erential equation a ( t ) y °° + b ( t ) y ° + c ( t ) y = f ( t ); where a ( t ), b ( t ), c ( t ), and f ( t ) are continuous functions. We found before that when a ( t ), b ( t ), and c ( t ) are constant functions, and f ( t ) is the zero function, we have the general solution y 0 ( t )= c 1 y 1 ( t )+ c 2 y 2 ( t ) where c 1 and c 2 are constants; { y 1 ,y 2 } is a fundamental set of solutions in terms of the functions y 1 ( t )= ° e λt cosh μt if b 2 4 ac> 0, e λt cos μt if b 2 4 ac< 0. y 2 ( t )= ° e λt sinh μt if b 2 4 ac> 0, e λt sin μt if b 2 4 ac< 0; and we have the constants λ = b 2 a and μ = ± | b 2 4 ac | 2 a . More generally, in order to solve the nonhomogeneous equation a ( t ) y °° + b ( t ) y ° + c ( t ) y = f ( t ) we assume the following: i. A fundamental set of solutions { y 1 ,y 2 } is known for the homogeneous equation a ( t ) y °° + b ( t ) y ° + c ( t ) y =0 . ii. The general solution is in the form y ( t )= u 1 ( t ) y 1 ( t )+ u 2 ( t ) y 2 ( t ) for some functions u 1 = u 1 ( t ) and u 2 = u 2 ( t ) to be found. Example. Consider the di±erential equation y °° 3 y ° 4 y =2 e t . We found above that the general solution is
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Unformatted text preview: y ( t ) = c 1 e t + c 2 e 4 t + 2 5 t e t = c 1 2 5 t e t + c 2 e 4 t . General Method. We explain how to Fnd the general solution of the nonhomogeneous equation a ( t ) y + b ( t ) y + c ( t ) y = f ( t ) . Perform the following steps: #1. ind a fundamental set of solutions { y 1 , y 2 } to the homogeneous dierential equation a ( t ) y + b ( t ) y + c ( t ) y = 0 . #2. Compute the integrals u ( t ) = t f ( ) a ( ) y 2 ( ) W ( ) d + c 1 and u 2 ( t ) = t f ( ) a ( ) y 1 ( ) W ( ) d + c 2 in terms of the Wronskian W ( t ) = y 1 ( t ) y 2 ( t ) y 1 ( t ) y 2 ( t ) . #3. orm the function y ( t ) = u 1 ( t ) y 1 ( t ) + u 2 ( t ) y 2 ( t ) . This is the general solution to the nonhomogeneous equation....
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