undefined Solution To derive: The formula W ( y 1 , y 2 ) = ( y 2 y 1 ) ′ y 1 2 ( y 1 ≠ 0 ) or − ( y 1 y 1 ) ′ y 2 2 ( y 2 ≠ 0 ) from W ( y 1 , y 2 ) = y 1 y ′ 2 − y 2 y ′ 1 . Formula used: Wronskian: The Wronskian W ( y 1 , y 2 ) is the determinant | y 1 y 2 y ′ 1 y ′ 2 | . That is, ( 6 ) W ( y 1 , y 2 ) = | y 1 y 2 y ′ 1 y ′ 2 | = y 1 y ′ 2 − y 2 y ′ 1 . Calculation: The equation (6) is, W ( y 1 , y 2 ) = y 1 y ′ 2 − y 2 y ′ 1 . Now derive ( 6 * ) from equation (6) as shown below. Consider the derivative ( y 2 y 1 ) ′ and obtain the first form as shown below. ( y 2 y 1 ) ′ = ( y ′ 2 y 1 − y 2 y ′ 1 y 1 2 ) ( By quotient rule ) ( y 2 y 1 ) ′ ⋅ y 1 2 = ( y ′ 2 y 1 − y 2 y ′ 1 y 1 2 ) ⋅ y 1 2 = y ′ 2 y 1 − y 2 y ′ 1 = W ( y 1 , y 2 ) Therefore, the first form W ( y 1 , y 2 ) = ( y 2 y 1 ) ′ y 1 2 ( y 1 ≠ 0 ) is derived. Now consider the derivative ( y 1 y 2 ) ′ and obtain the second form as shown below. ( y 1 y 2 ) ′ = ( y ′ 1 y 2 − y 1 y ′ 2 y 2 2 ) ( By quotient rule ) ( y 2 y 1 ) ′ ⋅ y 1 2 = ( y ′ 1 y 2 − y 1 y ′ 2 y 1 2 ) ⋅ y 1 2 − ( y 2 y 1 ) ′ ⋅ y 1 2 = − ( y ′ 1 y 2 − y 1 y ′ 2 y 1 2 ) ⋅ y 1 2 = y ′ 2 y 1 − y 2 y ′ 1 = W ( y 1 , y 2 ) Hence the second form W ( y 1 , y 2 ) = ( y 2 y 1 ) ′ y 1 2 ( y 1 ≠ 0 ) is also derived.

10th Edition

ISBN: 9781119664697

Author: Kreyszig

Publisher: WILEY

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ADVANCED ENGINEERING MATHEMATICS

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