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April 20th, 2015, 08:43 AM  #1 
Senior Member Joined: Jan 2015 From: usa Posts: 104 Thanks: 1  diff equation
hi can someone please help me solve this problem: a) determine all the functions y: ℝ → ℝ which satisfy y '' (x) 5y '(x) + 6y (x) = x, x ∈ ℝ b) determine all sequences (xn) n> = 0 which satisfy x{n+2}5x{n+1} +6x{n} = n , n∈N here's what I've done: a) homogeneous solutions are of the form y (x) = + λe2x μe3x , λ, μ ∈ ℝ we look for a particular soulution of the form y (x) = a + bx Differentiating this solution and by using the diff equation we find y (x) = 1 / 6x + 5/36 so the general solution is y (x) = + λe2x μe3x + 1 / 6x + 5/36 but I can not answer the question b) can someone Please help.thanks in advance. 
April 20th, 2015, 09:47 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,689 Thanks: 2669 Math Focus: Mainly analysis and algebra 
The homogeneous solution is $A2^n + B3^n$ and you can try $a + bn$ for a particular solution.

April 20th, 2015, 10:04 AM  #3 
Senior Member Joined: Jan 2015 From: usa Posts: 104 Thanks: 1  equ diff
hi thanks for your help can you please tell me how did you find that the homogenuous solution is A2^n + B3^n thanks 
April 20th, 2015, 10:13 AM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,689 Thanks: 2669 Math Focus: Mainly analysis and algebra 
I looked at your exponential terms for the differential equation! More seriously, the roots of the characteristic polynomial are the exponents in the solution to the differential equation and the exponential bases in the solution to the difference equation. 

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