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lecture_20 (dragged) 1 - 2 MA 36600 LECTURE NOTES WEDNESDAY...

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2 MA 36600 LECTURE NOTES: WEDNESDAY, MARCH 4 #1. Find 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 ) + c 1 and u 2 ( t )= ° t f ( τ ) a ( τ ) y 1 ( τ ) W ( τ ) + c 2 in terms of the Wronskian W ( t )= y 1 ( t ) y ° 2 ( t ) y ° 1 ( t ) y 2 ( t ) . #3. Form 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. This method is called Variation of Parameters . We remark that we can always write y ( t )= u 1 ( t ) y 1 ( t )+ u 2 ( t ) y 2 ( t )= c 1 y 1 ( t )+ c 2 y 2 ( t )+ Y ( t ) in terms of the function Y ( t )= ° t f ( τ ) a ( τ ) y 1 ( τ ) y 2 ( t ) y 1 ( t ) y 2 ( τ ) y 1 ( τ ) y ° 2 ( τ ) y ° 1 ( τ ) y 2 ( τ ) dτ. This function is a particular solution to the nonhomogeneous equation Note that when f ( t )i sth ez e ro function, we ²nd that u 1 ( t )= c 1 ,u 2 ( t )= c 2 ,Y ( t )=0 = y ( t )= c 1 y 1 ( t )+ c 2 y 2 ( t )
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