Question: find a general solution of homogeneous linear difference equation of the 2nd order with constant coefficients. How to solve it? Thank you. (pictures below)

task:

this should be solution:

Thank you!

Answer 1

To find the general solution of a homogeneous linear difference equation of the 2nd order with constant coefficients, you can use the characteristic equation method. Here are the steps:

  1. Write down the given difference equation in the form: ( a_n x_{n+2} + b_n x_{n+1} + c_n x_n = 0 ).

  2. Assume the solution is of the form ( x_n = r^n ).

  3. Substitute ( x_n = r^n ) into the difference equation and simplify to obtain the characteristic equation.

  4. Solve the characteristic equation to find the roots ( r_1 ) and ( r_2 ).

  5. Depending on the nature of the roots:

    • If ( r_1 ) and ( r_2 ) are distinct real roots, the general solution is ( x_n = A r_1^n + B r_2^n ), where ( A ) and ( B ) are constants determined by initial conditions.
    • If ( r_1 ) and ( r_2 ) are complex roots ( \alpha \pm \beta i ), the general solution is ( x_n = A (\alpha + \beta i)^n + B (\alpha - \beta i)^n ), where ( A ) and ( B ) are constants determined by initial conditions.
    • If ( r_1 = r_2 = r ) is a repeated real root, the general solution is ( x_n = (A + Bn) r^n ), where ( A ) and ( B ) are constants determined by initial conditions.
  6. Use initial conditions, if provided, to determine the values of constants ( A ) and ( B ) in the general solution.

  7. Write down the final general solution.

That's how you solve a homogeneous linear difference equation of the 2nd order with constant coefficients.

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Answer 2

# a_n = 4^n(An+b) #

We seek a solution to the Linear difference equation:

# a_(n+2) -8_(n+1) + 16a_n = 0 # ..... [A]

This is a Second Order Homogeneous linear difference equations, and is solved in a similar way to a second order linear differential equations with constant coefficients, by forming the Auxiliary Equation, which is the polynomial equation with the coefficients of the difference terms:

So, the associated Auxiliary equation is:

# m^2-8m+16 = 0#
# :. (m-4)^2 = 0 #

And so we have the solutions:

# m = +-4 # (repeated real root)

The roots of the auxiliary equation determine parts of the solution, which if linearly independent then the superposition of the solutions form the full general solution.

Thus the General Solution of the homogeneous equation [A] is:

# a_n = 4^n(An+b) \ \ \ \ # QED
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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