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362_pdfsam_math 54 differential equation solutions odd

362_pdfsam_math 54 differential equation solutions odd -...

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Chapter 6 Equation (6.5) will equal zero and, therefore, the differential equation will be satisfied for r = ± 1 and r = 2. Thus, three solutions to the differential equation are y = x , y = x 1 , and y = x 2 . Since these functions are linearly independent, they form a fundamental solution set. (b) Let y ( x ) = x r . In addition to (6.4), we need the fourth derivative of y ( x ). y (4) = ( y ) = r ( r 1)( r 2)( r 3) x r 4 = ( r 4 6 r 3 + 11 r 2 6 r ) x r 4 . Thus, if y = x r is a solution to this fourth order Cauchy-Euler equation, then we must have x 4 ( r 4 6 r 3 + 11 r 2 6 r ) x r 4 + 6 x 3 ( r 3 3 r 2 + 2 r ) x r 3 +2 x 2 ( r 2 r ) x r 2 4 xrx r 1 + 4 x r = 0 ( r 4 6 r 3 + 11 r 2 6 r ) x r + 6( r 3 3 r 2 + 2 r ) x r + 2( r 2 r ) x r 4 rx r + 4 x r = 0 ( r 4 5 r 2 + 4) x r = 0 . (6.6) Therefore, in order for y = x r to be a solution to the equation with x > 0, we must have r 4 5 r 2 + 4 = 0. Factoring this equation yields r 4 5 r 2 + 4 = ( r 2 4)( r 2 1) = ( r 2)( r + 2)( r 1)( r + 1) = 0 . Equation (6.6) will be satisfied if
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