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Unformatted text preview: Math 334 Lecture #15 4.2: Homogeneous n th-order ODEs with Constant Coefficients A solution of the homogeneous linear ODE L [ y ] = a y ( n ) + a 1 y ( n- 1) + + a n- 1 y + a n y = 0 has the form y = e rt if and only if r is a root of the characteristic equation a r n + a 1 r n- 1 + + a n- 1 r + a n = 0 . Distinct Real Roots. If the n roots r 1 , r 2 , . . . , r n of the characteristic equation are real and distinct (no two are equal), then the general solution of L [ y ] = 0 is y = c 1 e r 1 t + c 2 e r 2 t + + c n e r n t . [It can be shown that the Wronskian of the n solutions is nonzero for all t in R .] Example. Find a general solution of y 000 + 2 y 00- y- 2 y = 0 . The characteristic equation is r 3 + 2 r 2- r- 2 = 0 r 2 ( r + 2)- ( r + 2) = 0 ( r 2- 1)( r + 2) = 0 . The roots are r = 1 ,- 1 ,- 2, so that the general solution is y = c 1 e t + c 2 e- t + c 3 e- 2 t ....
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