Second Order Difference Equations « Complex Math Simple Life

Second Order Difference Equations

Posted in complex, difference, equation by beauangel on the February 19, 2007

Standard Form

$a_0y_t + a_1y_{t-1} + a_2y_{t-2} = f(t)$ or $a_0y_{t+2} + a_1y_{t+1} + a_2y_t = f(t)$

The solving of a second order difference equation is very similar to the method of solving a second order differential equation, which is discussed in this previous post here.

The General Solution of a second order difference equation has a Complementary Function and a Particular Solution.

The Complementary Function (CF) is found by writing the auxiliary equation

$a_0m^2 + a_1m + a_2 = 0$ and solving it to find the two roots of such a quadratic equation.

When there are two different roots, m₁ and m₂, the CF is written as
$y_t = A\left( m^t_1 \right) + B \left( m^t_2 \right)$ where A and B are arbitrary constants.
When there are two equal roots, such that m₁ = m₂ = m, then the CF is written as
$y_t = (A + Bt)m^t$ .
When there are two complex roots, $u + iv$ and $u - iv$ , then the CF is

$y_t = r^t (A \cos t\theta + B \sin t\theta)$ where $r = \sqrt{u^2 + v^2}$ and $\tan \theta = \frac{v}{u}$ .

Based on the standard form of the second order difference equation, the Particular Solution depends on the form $f(t)$ , the function in t on the RHS of the difference equation.

Again, similar to the case of differential equations, the Particular Solution (PS) is the same form as $f(t)$ but contains undetermined coefficients which are determined by comparing them with the function in the RHS of the given difference equation.

If the function is a constant, say c, then the general form of the PS is k, an unknown constant.

If the function is in the form $c \left( a^t \right)$ , then the general form of the PS is $k\left( a^t \right)$ .

If the function is in the form $at+ b$ , then a general linear function in t, $A_1t + A_2$ , is the general form of the PS.

A homogeneous second order difference equation has zero on the RHS. Its standard form is $a_1y_{t+2} + a_2y_{t+1} + a_3y_t = 0$ .

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romeo said,

on May 6, 2009 on 12:44 am

first orderhomogeneous difference equation

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