diff equation

[mona123]

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 (x_n) _{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) = + λe^2x μe^3x
, 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 _{/ 6}x + 5_/36
so the general solution is y (x) = + λe^2x μe^3x + 1 _{/ 6}x + 5_/36
but I can not answer the question b) can someone Please help.thanks in advance.

 

[Ishuda]

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 (x_n) _{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) = + λe^2x μe^3x
, 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 _{/ 6}x + 5_/36
so the general solution is y (x) = + λe^2x μe^3x + 1 _{/ 6}x + 5_/36
but I can not answer the question b) can someone Please help.thanks in advance.

For (a) you have the proper solution although I would write it differently as
y (x) =λe^2x + μe^3x + (1/6) x + 5/36
but I think we both mean the same thing.

For (b), difference equations of this type are in the same category as differential equations. You just have to learn the language, but basically, you treat them about the same. Assume a solution of the form of A xⁿ for the homogenous solution to get a quadratic equation in x and a homogenous solution of λ x₀ⁿ + μ x₁ⁿ where x₀ and x₁ are different solutions to the quadratic equation. Try to get the particular solution by using a quadratic
x_n = a + b n + c n²
and determine the coefficients a, b, and c [in this problem, the c coefficient will be zero as it was for the differential equation].

For further reading you might want to look at
https://www.cl.cam.ac.uk/teaching/2003/Probability/prob07.pdf