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An in-depth explanation of how to solve first-order linear difference equations, focusing on the explicit solution as a sum of particular and complementary solutions. topics such as the general first-order linear equation, homogeneous and inhomogeneous solutions, and the method of undetermined coefficients.
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Peter J. Hammond
Latest revision 2020 September 24th, typeset from dynEqLects20A.tex
Introduction: Difference vs. Differential Equations First-Order Difference Equations First-Order Linear Difference Equations: Introduction General First-Order Linear Equation Particular, General, and Complementary Solutions Explicit Solution as a Sum Constant and Undetermined Coefficients Stationary States and Stability for Linear First-Order Equations Local Stability of Nonlinear First-Order Equations
Athletics rules limit a walking step to be no longer than a stride. So a walking process that starts with the left foot might be described by the two coupled equations `m =
{ λ(r m− 1 )^ if^ m^ is odd `m− 1 if m is even and^ rm^ =
{ ρ(m− 1 )^ if^ m^ is even rm− 1 if m is odd for m = 0, 1 , 2 ,.... Or, if the length and direction of each pace are affected by the length and direction of its predecessor, bym =
{ λ(r m− 1 , m− 2 )^ if^ m^ is oddm− 1 if m is even and rm =
{ ρ(` m− 1 ,^ rm− 2 )^ if^ m^ is even rm− 1 if m is odd for University of Warwick, EC9A0 Maths for Economists, Day 7 m = 0, 1 , 2 ,.... Peter J. Hammond 4 of 54
Newtonian physics implies that a walker’s centre of mass must be a continuous function of time, described by a 3-vector valued mapping R+ 3 t 7 → (x(t), y (t), z(t)) ∈ R^3. The time domain is therefore T := R+. The same will be true for the position of, for instance, the extreme end of the walker’s left big toe. Newtonian physics requires that the acceleration 3-vector described by the second derivative (^) ddt^22 (x(t), y (t), z(t)) ∈ R^3 should be well defined for all t. The biology of survival requires it to be bounded. Actually, the motion becomes seriously uncomfortable unless the acceleration (or deceleration) is continuous — as my driving instructor taught me more than 50 years ago!
Let T = Z+ 3 t 7 → xt ∈ X describe a discrete time process, with X = R (or X = Rm) as the state space. Its difference at time t is defined as ∆xt := xt+1 − xt A standard first-order difference equation takes the form xt+1 − xt = ∆xt = dt (xt ) where each dt : X → X , or equivalently, T × X 3 (t, x) 7 → dt (x)
Obviously, the difference equation xt+1 − xt = ∆xt = dt (xt ) is equivalent to the recurrence relation xt+1 = rt (xt ) where T × X 3 (t, x) 7 → rt (x) = x + dt (x), or equivalently, dt (x) = rt (x) − x. Thus difference equations and recurrence relations are entirely equivalent. We follow standard mathematical practice in using the notation for recurrence relations, even when discussing difference equations. We may write “difference equation” even when considering a recurrence relation.
Introduction: Difference vs. Differential Equations First-Order Difference Equations First-Order Linear Difference Equations: Introduction General First-Order Linear Equation Particular, General, and Complementary Solutions Explicit Solution as a Sum Constant and Undetermined Coefficients Stationary States and Stability for Linear First-Order Equations Local Stability of Nonlinear First-Order Equations
Consider a consumer who, in discrete time t = 0, 1 , 2 ,... : I (^) starts each period t with an amount wt of accumulated wealth; I (^) receives income yt ; I (^) spends an amount et ; I (^) earns interest on the residual wealth wt + yt − et at the rate rt. The process of wealth accumulation is then described by any of the equivalent equations wt+1 = (1 + rt )(wt + yt − et ) = ρt (wt − xt ) = ρt (wt + st ) where, at each time t, I (^) ρt := 1 + rt is the interest factor; I (^) xt = et − yt denotes net expenditure; I (^) st = yt − et = −xt denotes net saving.
We transform the difference equation wt+1 = ρt (wt − xt ) by using the compound interest factor Rt = ∏t k−=0^1 ρk in order to discount both future wealth and expenditure. To do so, define new variables ωt , ξt for the present discounted values (PDVs) of, respectively:
Introduction: Difference vs. Differential Equations First-Order Difference Equations First-Order Linear Difference Equations: Introduction General First-Order Linear Equation Particular, General, and Complementary Solutions Explicit Solution as a Sum Constant and Undetermined Coefficients Stationary States and Stability for Linear First-Order Equations Local Stability of Nonlinear First-Order Equations
The matrix form of the difference equation is Cx = f, where:
cst =
−as if t = s 1 if t = s + 1 0 otherwise for s = 1, 2 ,... , T and t = 0, 1 , 2 ,... , T ;
The matrix equation Cx = f can be written in partitioned form as (U eT ) (xT^ −^1 xT
= f where:
Alternatively, a terminal condition for the difference equation xt − xt− 1 = ft specifies an exogenous value ¯xT for the value xT at the terminal time T. It leads to a unique solution as a backward sum xt = ¯xT − ∑T s=0^ −t −^1 fT −s of the exogenously specified I (^) terminal state ¯xT ; I (^) preceding backward differences −fT −s (s = 0, 1 ,... , T − t − 1).
Introduction: Difference vs. Differential Equations First-Order Difference Equations First-Order Linear Difference Equations: Introduction General First-Order Linear Equation Particular, General, and Complementary Solutions Explicit Solution as a Sum Constant and Undetermined Coefficients Stationary States and Stability for Linear First-Order Equations Local Stability of Nonlinear First-Order Equations
