Explanation Given: The given differential equation is y ‴ − 2 y ″ + y ′ = x . Approach: To determine the particular solution of the equation first write down the auxiliary equation. Then find out the roots of the auxiliary equation. Then find the complimentary function of the given equation. Then find out four Wronskian of determinants. Use the formula for finding the particular solution of the equation. y p ( x ) = ∑ k = 1 n y k ( x ) ∫ g ( x ) W K ( x ) W [ y 1 , ... , y n ] ( x ) d x Calculation: To determine the particular solution of the equation first write down the auxiliary equation. y ‴ − 2 y ″ + y ′ = x ... ( 1 ) The auxiliary equation is r 3 − 2 r 2 + r = 0 ... ( 2 ) To find factor of the equation put r = 0 in the equation. Now equation become- ( 0 ) 3 − 2 ( 0 ) 2 + 0 = 0 0 = 0 Thus, r = 0 is a factor of the equation. Now, put r in the equation to find the other factors- r 3 − 2 r 2 + r = 0 r ( r 2 − 2 r + 1 ) = 0 r ( r − 1 ) ( r − 1 ) = 0 r = 0 , 1 , 1 The complimentary function is- C F = c 1 e 0 x + c 2 e x + c 3 x e x Compare with C F = c 1 y 1 ( x ) + c 2 y 2 ( x ) + c 3 y 3 ( x ) . Now y 1 ( x ) = 1 , y 2 ( x ) = e x and y 3 ( x ) = x e x Write four wronskian of determinant. W [ y 1 , y 2 , y 3 ] ( x ) = | y 1 y 2 y 3 y ′ 1 y ′ 2 y ′ 3 y ″ 1 y ″ 2 y ″ 3 | W [ y 1 , y 2 , y 3 ] ( x ) = | 1 e x x e x 0 e x ( x + 1 ) e x 0 e x ( x + 2 ) e x | Now on solving. W [ y 1 , y 2 , y 3 ] ( x ) = e 2 x Now find the value of W 1 , W 2 and W 3 W 1 ( x ) = ( − 1 ) 3 − 1 W [ y 2 , y 3 , ] ( x ) = ( − 1 ) 2 | e x x e x e x ( x + 1 ) e x | Write common factor from each of the column. W 1 ( x ) = ( − 1 ) 3 − 1 W [ y 2 , y 3 , ] ( x ) = e x ⋅ e x | 1 x 1 ( x + 1 ) | W 1 ( x ) = ( − 1 ) 3 − 1 W [ y 2 , y 3 , ] ( x ) = e 2 x Similarly for W 2 follow same procedure. W 2 ( x ) = ( − 1 ) 3 − 2 | 1 x e x 0 ( x + 1 ) e x | Take common factor from the column

7th Edition

ISBN: 9780135902738

Author: Nagle

Publisher: PEARSON

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Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

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