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 find the general solution to the linear second order di ff 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 λ t cosh μt if b 2 4 a c > 0, e λ t cos μt if b 2 4 a c < 0. y 2 ( t ) = e λ t sinh μt if b 2 4 a c > 0, e λ t sin μt if b 2 4 a c < 0; and we have the constants λ = b 2 a and μ = | b 2 4 a c | 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 ff 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 − 2 5 t e − t ³ = ´ c 1 − 2 5 t µ 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. ²ind a fundamental set of solutions { y 1 , y 2 } to the homogeneous di±erential equation a ( t ) y °° + b ( t ) y ° + c ( t ) y = 0 . #2. Compute the integrals u ( t ) = − ¶ t f ( τ ) a ( τ ) y 2 ( τ ) W ( τ ) dτ + c 1 and u 2 ( t ) = ¶ t f ( τ ) a ( τ ) y 1 ( τ ) W ( τ ) dτ + c 2 in terms of the Wronskian W ( t ) = y 1 ( t ) y ° 2 ( t ) − y ° 1 ( t ) y 2 ( t ) . #3. ²orm 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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