Differential Equations

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(1) General Differential Equations
(2) Dynamical Systems of Differential Equations
(A) Homogeneous System
(B) Typical System

(1) General Differential Equations

Economists use differential equations largely in the context of dynamical systems, i.e. in systems where time, t, is one of the variables. However, differential equations are defined more generally than this. In this section, we provide general definitions and revert only to including time as the explicit variable in the next section.

Ordinary Differential Equation: an ordinary differential equation of nth order is the following implicit relationship:

ｦ (x, y, y｢ , .., y⁽ⁿ⁾) = 0

where x is a variable, y is an unknown function of x and y｢ , y｢ ｢ , .., y⁽ⁿ⁾ are the n derivatives of y.

Intuitively, a differential equation is an equation involving derivatives of an unknown function y. The problem is one of finding this function - thus a solution to a differential equation is a function y = j (x) which satisfies:

ｦ (x, j (x), j ｢ (x), .., j ⁽ⁿ⁾(x)) = 0

For conditions establishing the existence of a solution, j (x), the Cauchy-Peano theorem, we refer to any text on this matter and shall thus pass it over in silence here.

We shall concern ourselves throughout with first order differential equations (FODE), so that we have:

ｦ (x, y, y｢ )

As x is included explicitly, then this is a "non-autonomous" system; if x were excluded, then we have an autonomous system. We can convert from non-autonomous to autonomous systems via a change of variable technique which we shall not pursue here. We shall also focus the bulk of our attention on linear FODE. This is defined as follows:

Linear FODE: a differential equation is a linear first order differential equation if it can be written in the form:

a(x)y｢ + b(x)y = c(x)

where a, b, c are functions of x, where a(x), b(x) are referred to as "coefficients" and c(x) is referred to as the "second member".

Theorem: The general solution of a linear FODE is the sum of the particular solution of the complete equation, a(x)y｢ +b(x)y = c(x) and the solution of the equation without second member, a(x)y｢ +b(x)y = 0.

Proof: let y₀ be a particular solution to the complete equation. Let y be the general solution such that y = y₀ + z. We must prove that z is a solution to the equation without the second member. As y solves a(x)y｢ + b(x)y = c(x), then a(x)[y₀｢ + z｢ ] + b(x)[y₀ + z] = c(x), or:

a(x)y_o｢ + a(x)z｢ + b(x)y₀ + b(x)z = c(x)

Since y₀ is a particular solution to the complete equation, the a(x)y₀｢ + b(x)y₀ - c(x) = 0, thus the previous equation reduces to:

a(x)z｢ + b(x)z = 0

thus z is a solution to the equation without the second member.ｧ

Bernoulli Equation: A differential equation is called a Bernoulli equation if it can be written in the form:

y｢ + a(x)y = b(x)y^m

where a, b are functions of x and m is a constant (m ｹ 0, m ｹ 1).

Resolution: if y ｹ 0, rewrite the Bernoulli equation as:

y｢ /y^m + a(x)/y^m-1 = b(x)

and let z = 1/y^m-1 and z｢ = (1-m)y｢ /y^m so y｢ /y^m = z｢ /(1-m). Thus, rearraning:

z｢ /(1-m) + a(x)z = b(x)

which is a linear FODE we can solve.

Ricatti Equation: A differential equation is called a Riccati equation if it can be written in the form:

y｢ = a(x)y² + b(x)y + c(x)

where a, b, c are functions of x.

Resolution: Let y₁ be a particular solution of the Riccati equation. Then, setting y = y₁ + z, then this becomes:

y₁｢ + z｢ = a(x)(y₁+z)² + b(x)(y₁ + z) + c(x)

since y₁ is a particular solution then we obtain y₁｢ - a(x)y₁² - b(x)y₁ - c(x) = 0, so, after some algebra, the previous equation becomes:

z｢ = a(x)z² + 2[a(x)y₁ + b(x)]z

which is a Bernoulli equation for m = 2, which we can solve.

(2) Dynamical Systems of Differential Equations

In our previous section, we defined a differential equation as a general function. Now, we shall consider time explicitly and thus consider differential equations ｦ (t, x, x｢ , .., x⁽ⁿ⁾) where, note, time, t ﾎ R₊, is now the variable and x(t) is a function of time (and x｢ , x｢ ｢ , etc. are its first and higher order derivatives). We shall in this section focus our attention exclusively on systems of linear first order differential equations. This translates effectively to a system of n differential equations of the following form:

dx₁(t)/dt = a₁₁x₁(t) + a₁₂x₂(t) + ..... a_1nx_n(t) + b₁(t)

dx₂(t)/dt = a₂₁x₁(t) + a₂₂x₂(t) + ..... a_2nx_n(t) + b₂(t)

............................................................................

dx_n(t)/dt = a_n1x₁(t) + a_n2x₂(t) + ..... a_nnx_n(t) + b_n(t)

or, letting x｢ (t) = [dx₁(t)/dt, dx₂(t)/dt, ... dx_n(t)/dt]｢ , x(t) = [x₁(t), x₂(t), .... x_n(t)]｢ , b(t) = [b₁(t), b₂(t), ..., b_n(t)]｢ , and letting:

be a matrix of (constant) coefficients, then the system can be rewritten as:

x｢ (t) = Ax(t) + b(t)

Throughout the following, the term t will be dropped as an argument of x｢ (t) and x(t) if no confusion is risked.

(A) The Homogeneous System

If b(t) = 0, then x｢ (t) = Ax(t) is homogeneous. The solution to a homogenous system can be expressed as follows:

Theorem: Let x｢ = Ax is a homogeneous linear first-order system. If x = ve^lt is a solution to this system (where v = [v₁, v₂, ..., v_n]｢ ], then l is an eigenvalue of A and v is the corresponding eigenvector.

Proof: If x = ve^lt, then x｢ = l ve^lt and thus substituting for x and x｢ , the homogeneous system can be rewritten as l ve^lt = Ave^lt, which, dividing through by e^lt, yields us the eigenvalue system l v = Av or (A - l I)v = 0. In other words, for a non-trivial solution, it must be that |A-l I| = 0, which is the characteristic equation of matrix A. Thus, l is an eigenvalue of A and v is its associated eigenvector.ｧ

As the matrix A has n eigenvalues, l ₁, .., l _n and n associated eigenvectors, v₁, v₂, .., v_n, then each term v_ie^lit is a solution to the homogeneous system x｢ = Ax. The following theorem establishes that any linear combination of these terms are also solutions to x｢ = Ax:

Theorem: if A is a real n ｴ n matrix with n distinct eigenvalues, l ₁, .., l _n and associated eigenvectors, v₁, v₂, ..., v_n, then z(t) = ・/font> _i=1ⁿc_iv_ie^lit is also a solution to the homogeneous system x｢ = Ax where c₁, .., c_n are arbitrary, possibly complex, constants.

Proof: We wish to prove that as v₁e^l1t, v₂e^l2t, .., v_ne^lnt are all independent solutions to the system x｢ = Ax, then so is their linear combination z(t) = ・/font> _i=1ⁿ c_iv_ie^lit. This is easily noticed as, taking first derivatives of z(t), we obtain z｢ (t) = ・/font> _ic_il _iv_i e^lit which as l _iv_i = Av_i, then z｢ (t) = ・/font> _ic_iAv_ie^lit = Az(t) by the definition of z(t). Thus, z(t) is a solution to the system x｢ = Ax.ｧ

The matrix F (t) = [v₁e^l1t, v₂e^l2t, .., v_ne^lnt] is sometimes referred to as the "fundmental matrix" as v_ie^lit are linearly independent of each other (a result of l ₁, l ₂, .., l _n being distinct eigenvalues). This implies that any solution x(t) to the system x｢ = Ax can be expressed as a unique combination of the vectors in the fundamental matrix. (we omit the proof). Consequently, what is commonly referred to as the general solution to the system x｢ = Ax is given as:

x(t) = ・/font> _i=1ⁿ c_iv_ie^lit

where, as noted earlier, c₁, .., c_n are arbitrary, possibly complex, constants. If the eigenvalues are not dinstinct, things get a bit complicated but nonetheless, as repeated roots are not robust, or "structurally unstable" (i.e. do not survive small changes in the coefficients of A), then these can be generally ignored for practical purposes (cf. Murata, 1977).

Let us now turn to another interesting issue. Recall that a matrix A is "diagonalizable" if there is a matrix, P, such that P^-1AP is a diagonal matrix. We now turn to the following:

Theorem: An n-square matrix is diagonalizable if and only if it has n independent eigenvectors.

Proof: Define the modal matrix P = [v₁, v₂, .., v_n], thus P is a (n ｴ n) matrix whose n columns are n eigenvectors of A. Thus, as Av_i = l _iv_i for i = 1, .. n, then A[v₁, v₂, .., v_n] = [l ₁v₁, l ₂v₂, .., l _nv_n], or simply AP = PL where L is a diagonal matrix with the eigenvalues l ₁, l ₂, .., l _n of A arrayed along the diagonal, i.e.

	l 1	0	....	0
L =	0	l 2	....
	....	....	....	....
	0	0	....	l n

As AP = PL , then obviously P^-1AP = L , thus the matrix P diagonalizes A. For P^-1 to exist, the columns of P, i.e. the eigenvectors v_i, must be linearly independent. Conversely, if P is non-singular, P^-1 exists and P^-1AP = L , i.e. P diagonalizes A.ｧ

For the next set of theorems, it is worth noting that Taylor's expansion of the function ｦ (t) = e^at around t = 0 is:

ｦ (t) = e^at = 1 + at/1! + a²t²/2! + a³t³/3! + ....

As a consequence, the following theorem can be stated:

Theorem: The solution of x｢ (t) = Ax(t), x(0) = x₀ is x(t) = e^Atx₀.

Proof: Taylor's expansion of x(t) around t = 0 yields:

x(t) = x(0) + x｢ (0)t/1! + x｢ ｢ (0)t²/2! + x｢ ｢ ｢ (0)t³/3! + ....

As x｢ (t) = Ax(t), then x｢ ｢ (t) = Ax｢ (t) = AAx(t) = A²x(t). Similarly, x｢ ｢ ｢ (t) = A³x(t) and so on. Thus, at t = 0, we have x｢ (0) = Ax(0) = Ax₀, x｢ ｢ (0) = A²x(0) = A²x₀, x｢ ｢ ｢ (0) = A³x(0) = A³x₀, etc. from the initial condition x(0) = x₀. Thus, replacing these in the Taylor's expansion:

x(t) = x₀ + Ax₀t/1! + A²x₀t²/2! + A³x₀t³/3! + ....

or, factoring out x₀:

x(t) = [I + At/1! + A²t²/2! + A³t³/3! + ....]x₀

where I is the identity matrix. But, as established earlier, we know that e^At = [I + At/1! + A²t²/2! + A³t³/3! + ....], so x(t) = e^Atx₀.ｧ

We can now turn to the following:

Theorem: The solution of x｢ (t) = Ax(t), x(0) = x₀, A diagonalizable is:

x(t) = e^Atx₀ = Pe^LtP^-1x₀

where P = [v₁, v₂, ..., v_n] is the modal matrix whose columns are eigenvectors of A and L is a diagonal matrix whose diagonal elements are distinct eigenvalues of A.

Proof: Distinct eigenvalues ensure linearly independent eigenvectors and hence non-singularity of P and, by our previous theorem, the diagonalizability of A. Thus, P^-1AP = L or A = PL P^-1. Thus, A² = AA = (PLP^-1)(PLP^-1) = PLIL P^-1 = PL ²P^-1. Similarly, A³ = PL³P^-1 and so on. Now, recall that:

e^At = [I + At/1! + A²t²/2! + A³t³/3! + ....]

so, substituting in for A, A², etc. and recalling that I = PP^-1, then:

e^At = [PP^-1 + (PL P^-1)t/1! + (PL ²P^-1)t²/2! + (PL ³P^-1) t³/3! + ....]

or factoring out P to the left and P^-1 to the right:

e^At = P[I + Lt/1! + L²t²/2! + L ³t³/3! + ....]P^-1

but, as we know by definition, e^Lt = [I + L t/1! + L ²t²/2! + L ³t³/3! + ....], thus this reduces to:

e^At = Pe^LtP^-1

hence:

x(t) = e^Atx₀ = Pe^LtP^-1x₀

as was to be shown.ｧ

Now, recall that the fundamental matrix was defined as F(t) = [v₁e^l1t, v₂e^l2t, .., v_ne^lnt] where each column is an independent solution of the homogeneous system, x｢ (t) = Ax(t). Also, recall that the general solution was:

x(t) = ・/font> _ic_iv_ie^lit

or, letting c = [c₁, .., c_n]:

x(t) = F (t)c

It is elementary to note, then, that F (t) = Pe^Lt by the definition of P and L . Thus:

x(t) = Pe^Ltc

But we also know that x(t) = Pe^LtP^-1x₀, thus it must be that c = P^-1x₀.

Thus, in short, a solution to the homogeneous system x｢ = Ax can be obtained by trying a solution x(t) = c₁v₁e^l1t + c₂v₂e^l2t +....+ c_nv_ne^lnt where l ₁, l ₂, ..., l _n are the eigenvalues of A, v₁, v₂, .., v_n are its eigenvectors and c₁, c₂, .., c_n the constants to be determined by the initial conditions.

(B) The Typical System

Let us now turn to a typical, non-homogeneous system of linear first order differential equations. Thus, turning away from the homogeneous case, we are now considering the system:

x｢ (t) = Ax(t) + b

where b ｹ 0 and, note, b is not a function of time. Consider now the following:

Theorem: The solution to x｢ = Ax + b with initial condition x(0) = x₀ is the following: x(t) = e^Atk - A^-1b where k = x₀ + A^-1b or, in alternative form, provided A is diagonalizable, x(t) = PeL ^tP^-1k - A^-1b.

Proof: Let y = x + A^-1b. Then, as b is independent of time, taking the time derivative, y｢ = x｢ . Thus, substituting, y｢ = Ax + b = Ax + AA^-1b = A(x + A^-1b) = Ay, i.e. we obtain a homogenous system y｢ = Ay. We know that the solution to a homogeneous system is y(t) = e^Aty₀ = PeL ^tP^-1y₀. For the first, note that y = e^Aty₀ implies x(t) + A^-1b = e^At[x(0) + A^-1b] or simply x(t) = e^At[x(0) + A^-1b] - A^-1b or, by the definition of k, x(t) = e^Atk - A^-1b. For the second, y(t) = PeL ^tP^-1y₀ implies x(t) + A^-1b = PeL ^tP^-1[x(0) + A^-1b] or x(t) = PeL ^tP^-1[x(0) + A^-1b] - A^-1b, or, once again, by definition of k, x(t) = PeL ^tP^-1k - A^-1b.ｧ

It can be noticed that the latter term x(t) = PeL ^tP^-1k - A^-1b can be expressed as:

x(t) = c₁v₁el ^1t + c₂v₂el ^2t +....+ c_nv_nel ^nt + x_p

or:

x(t) = F (t)c + x_p

where l ₁, l ₂, ..., l _n are the eigenvalues of A and v₁, v₂, .., v_n are its associated eigenvectors, so the fundamental matrix F (t) = [v₁el ^1t, v₂el ^2t, .., v_nel ^nt] = PeL ^t; the constants c₁, c₂, .., c_n are determined by the initial conditions, i.e. c = P^-1k = P^-1[x(0) + A^-1b]; and x_p is the particular integral (x_p = A^-1b).

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