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April 20th, 2015, 08:43 AM   #1
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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.
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April 20th, 2015, 09:47 AM   #2
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The homogeneous solution is $A2^n + B3^n$ and you can try $a + bn$ for a particular solution.
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April 20th, 2015, 10:04 AM   #3
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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
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April 20th, 2015, 10:13 AM   #4
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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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