lecture_14 (dragged) 2

lecture_14 (dragged) 2 - MA 36600 LECTURE NOTES: WEDNESDAY,...

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MA 36600 LECTURE NOTES: WEDNESDAY, FEBRUARY 18 3 Wronskian. We have seen that given the homogeneous equation L [ y ] = 0, we can fnd many solutions when given just a Few. However, when do we have all solutions? That is, iF y 1 = y 1 ( t ) and y 2 = y 2 ( t ) are solutions to L [ y ] = 0, when is y = c 1 y 1 + c 2 y 2 the general solution? We will show the Following: Consider the initial value problem a ( t ) y °° + b ( t ) y ° + c ( t ) y = 0; y ( t 0 )= y 0 ,y ° ( t 0 )= y ° 0 . Say that y 1 = y 1 ( t ) and y 2 = y 2 ( t ) are solutions to the di±erential equation (but not necessarily satisFying the initial conditions!) such that the quantity W 0 = y 1 ( t 0 ) y ° 2 ( t 0 ) y ° 1 ( t 0 ) y 2 ( t 0 )=det ° y 1 ( t 0 ) y 2 ( t 0 ) y ° 1 ( t 0 ) y ° 2 ( t 0 ) ± is nonzero. Then the solution to the initial value problem is the linear combination y ( t )= c 1 y 1 ( t )+ c 2 y 2 ( t ) in terms oF the constants c 1 = y 0 y ° 2 ( t 0 ) y ° 0 y 2 ( t 0 ) W 0 and c 2 = y 0 y ° 1 ( t 0 ) y ° 0 y 1 ( t 0 ) W 0 . We explain why this is true. By the Principle oF Superposition, the linear combination y = c 1 y 1 + c 2 y 2 is a solution to the di±erential equation. It suffices to explain how to compute the constants c
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