EC5555 Economics Masters Refresher Course in Mathematics September 2013

Transcription

1 EC5555 Economics Masters Refresher Course in Mathematics September 013 Lecture 9 Differential equations Francesco Feri

2 Integration and differential equations. Definition: a differential equation includes the derivative of a function as a variable to model the change in the level of a variable E.g. a or ax or ax These examples have onl first derivatives. The are called first order differential equations. Second order differential equations include second derivatives and so on. d a

3 Differential equations. Linear equations have no powers in x and no interaction terms between and x Homogeneous equations are ones such that if we multipl all variables b a constant, the differential equation is unchanged. x x or or a a ax ( ax) homogeneous non homogeneou s In the most common examples, x = time, usuall indicated b t In economics, decisions are rarel taken continuousl. Hence differential equations are usuall approximations to processes that might be more exactl represented via difference equations: a t1 t1 To solve a differential equation completel we also need information about at some time t. E.g. starting point or end point. or t a 3

4 Solving a simple homogeneous equation We are interested in finding the value of an underling variable at an point in time when we are onl given information on how the variable evolves over time (the differential) Example 1 f ( t) f ( t) 0 Integrate both sides wrt t x f ( t) f ( t) c Eg. f(t) = t 1 = / Gives x = t 3 /3 t + c So given t can work out the value of x in an period 4

5 Solving a simple homogeneous equation. a One wa to approach the solution is to treat and as separate items and integrate: 1 a 1 a ln( ) e at c c e at Ae at 5

6 Now (t) = Ae at Solving a simple homogeneous equation. (is an equation commonl used to describe the evolution of GDP over time where a = rate of growth) Note that A could be an value (in the growth literature A is used as a measure of technical progress) (t) = Ae at is said to be the general solution to the differential equation For a particular value of A this becomes a particular solution If we start off the process (0)=A, then this gives the definite solution Note the solution ((t) = Ae at ) is free of an derivative and is not a numerical value rather a function (or a time path ) giving the value of at an point in time 6

7 Solving a simple non-homogeneous equation a b The solution to the related homogeneous equation is called the complementar function ( c ) We call the particular integral ( p ) an particular solution to the nonhomogeneous equation) The sum of the complementar function and the particular integral constitutes the general solution of a 1 st order linear non-homogenous differential equation 7

8 Solving a simple non-homogeneous equation. (1) a b To solve proceed in steps: Look for an value that satisfies the equation the simplest is to let (t) = k = a constant then / = 0 and (1) becomes - a = b and (t) = -b/a = k (a 0 ) This solution to the non-homogeneous equation is called the particular integral 8

9 Solving a simple non-homogeneous equation. Step : We consider the homogeneous related equation a And we have alrea seen that the solution to this tpe of homogenous differential equation is (t) = Ae at 9

10 Solving a simple non-homogeneous equation The solution to the related homogeneous equation (complementar function ( c ) in this case is (t) = Ae at A particular solution to the non-homogeneous equation (particular integral ( p )) in this case is (t) = -b/a The sum of the complementar function and the particular integral constitutes the general solution of a 1 st order linear non-homogenous differential equation, in this case: (t) = c + p = Ae at - b/a to get the definite solution we need to impose an initial condition: the value of at t = 0 In this example (0) = Ae 0 b/a = A (b/a), A = (0) + (b/a) (t) = ((0) + (b/a) )e at (b/a) 10

11 Example: a=-1, b ==-K e t Often in economics the particular integral will represent the equilibrium of a sstem (here =), while the complementar function supplies the namics to get there (from the initial condition value) 11

12 Example Solving a simple non-homogeneous equation Solve / +(t) = 6 with an initial condition (0) = 10 The constant solution / = 0 gives the particular integral p = 6/ = 3 The complementar function is the solution to / = -(t) (with the constant set to zero) Or: 1 Integrating both sides wrt t: ln( ) c ln( ) 1 e t ( c t Ae e 1 k t t c ; k c c 1 c 1 ) 1

13 Solving a simple non-homogeneous equation So the general solution is: (t) = c + p = Ae -t + 3 The definite solution is to use the value for the initial conditions t=0, (0) =10 (0) = Ae 0 +3 = A + 3 so then A = (0) 3 = 10 3 = 7 So then (t) = [10 3]e -t + 3 = 7e -t + 3 This is saing the sstem converges to the value 3, starting from an initial value of 10 as t 13

14 Solving a simple non-homogeneous equation Sometimes solution is written down as v z ( t) e v A ze where A is an arbitrar constant e v A v is thecomplementar function e v ze v is the particular integral The particular integral p gives the inter-temporal equilibrium level of (t) The complementar function c gives the deviation from the equilibrium For (t) to be namicall stable c must approach 0 as t goes to infinit 14

15 Solving a simple non-homogeneous equation Eg: Find the general solution for / +4 =1 Let v = 4 and z = 1 Using ( t) e v ( t) e 4t ( t) Ae A 4t 4t 4t A 1e e A 3e 4t 3 ze v The 1 st term is the complementar function and the second is the particular integral As t goes to infinit the complementar function goes to zero and so (t) approaches the level of the particular integral Often helpful to check the solution b differentiating backwards 15

16 Exact differential equations a total differential of F(,t) is: df F F t If df = 0, the result is called an exact differential equation. The general solution should be of the form F = c (c is a constant) Usuall though we seek to find F given the differential. The statement that a differential equation of the form: (M and N are functions) is exact is equivalent to the statement that there exists F such that B Young s theorem: 0 M N M F ; N F t F t F t So F exists provided: 16 M t N

17 Solving exact differential equations M N t Step 1. Check that If it is then we have an exact equation. If it is not exact then we ma still be able to solve (see later) Step. Integrate M partiall with respect to. The result is: F M g(t) Note the term g(t) which we do not know et. Step 3. Partiall differentiate the result in with respect to t to get N: F M g'( t t t ) Note that g is the derivative of g with respect to t. Step 4. We know N so we can solve this equation to find g. From that we can integrate g to find g then use starting conditions to find the general solution. N 17

18 Solving exact differential equations - example Solve 4t Step 1. We rewrite the differential equation: 0 4t So M =1 and N =-4t. M t 0 N Step. Integrate M partiall with respect to. The result is: F Step 3. Partiall differentiate the result in with respect to t to get N: F t Step 4. find g. g =-4t, so integrate g to find g: So, M g( t) g( t) t F k g' ( t) t 0 g'( t) 4t or c t g' k or t t c 18

19 Integrating factors Sometimes M and N mean that the differential equation is inexact, but we can still find another function of t and which we can multipl through b to get an exact equation. M N Example. 0 t 1 t Multipl through b t to get: Step 1. Check M N t t 0 t t Step. Integrate M partiall with respect to. The result is: F M g( t) t g( t) Step 3. Partiall differentiate the result in with respect to t to get N: F M g' ( t) t g' N t t t Step 4. From this we get g = 0 or g = k, F t k or c t or c t 19