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# 5.5we - 5.5 EULERS HOMOGENEOUS EQUATIONS KIAM HEONG KWA A...

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5.5: EULER’S HOMOGENEOUS EQUATIONS KIAM HEONG KWA A homogeneous linear equation of the form (1) x n d n y dx n + α 1 x n - 1 d n - 1 y dx n - 1 + α 2 x n - 2 d n - 2 y dx n - 2 + · · · + α n y = 0 , where α i R for each i , i = 1 , 2 , · · · , n , is called an n th order Euler’s equation . By making the substitution (2) t = ln x in the case x > 0, it can be reduced to a homogeneous equation of the same order with constant coefficients. This follows from the fact that (3) dy dt = x dy dx , d 2 y dt 2 = x 2 d 2 y dx 2 + x dy dx , and, more generally, for every k N , (4) d k y dt k = x k d k y dx k + a k, 1 x k - 1 d k - 1 y dx k - 1 + a k, 2 x k - 2 d k - 2 y dx k - 2 + · · · + a k,k - 2 x 2 d 2 y dx 2 + x dy dx , for some a k,i R , i = 1 , 2 , · · · , k - 2. We shall construct the solution of the second-order Euler’s equation (5) x 2 d 2 y dx 2 + αx dy dx + βy = 0 , where α, β R , on the interval (0 , ) 1 . The solution on the interval ( -∞ , 0) can then be calculated by the substitution ζ = - x .

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