Chapter 8: Linear Differential Equations

8.1 Systems of linear differential equations with constant coefficients. In this chapter, we apply some of the results of chapter 7 to systems of linear differential equations. A system of linear differential equations of first order is

ì
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ï
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ï
ï
í
ï
ï
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ï
ï
î

dx1 dt

=

a₁₁x₁(t)+a₁₂x₂(t)+...+a_1nx_n(t)+f₁(t)

dx2 dt

=

a₂₁x₁(t)+a₂₂x₂(t)+...+a_2nx_n(t)+f₂(t)

¼

¼

dxn dt

=

a_n1x₁(t)+a_n2x₂(t)+...+a_nnx_n(t)+f(t)

(1)

Here a_ij, i,j = 1,2,...,n, are known coefficients, independent of t, f₁(t),...,f_n(t) are some known functions of t, and x₁(t),..., x_n(t) are unknown functions. A set of functions x₁(t), x₂(t),...,x_n(t) is called a solution of the system (1) if they are differentiable and when substituted into equations in (1) then the equations are satisfied.

Let us denote by A a square matrix whose elements are a_ij, i.e. A = [a_ij]_n×n. Further, let f be a vector-valued function, i.e a column function defined by

f(t) =

é ê ê ê ê ê ê ê ë

f1(t) f2(t) · · fn(t)

ù ú ú ú ú ú ú ú û

and x(t) be the unknown vector valued function defined by

x(t) =

é ê ê ê ê ê ê ê ë

x1(t) x2(t) · · xn(t)

ù ú ú ú ú ú ú ú û

Then, as can be seen easily (and similarly to the case of systems of linear algebraic equations), that system (?) can be rewritten in the following compact form

dx dt

= Ax(t)+f(t).

(2)

If f(t) º 0, then system (1) (or, what is the same, equation (2)) is called homogeneous. Otherwise, it is called inhomogeneous. The homogeneous equation

dx dt

= Ax(t).

(3)

always has a solution x(t) º 0. Such a solution of the homogeneous equation is called the trivial solution. The following theorem about the relationship between solutions of homogeneous and inhomogeneous equations is analogous to the corresponding theorem for linear algebraic equations, and the proof remains the same.

Theorem 1 Let x₀ be a particular solution of the inhomogeneous equation (2). Then for every solution x(t) of the homogeneous equation (3), the function

y(t) = x0(t)+x(t)

(4)

is a solution of the inhomogeneous equation (2). Conversely, if y(t) is an arbitrary solution of the inhomogeneous equation (2), then there exists a solution x(t) of the homogeneous equation (3), such that formula (4) holds.

From Theorem ? of chapter 7 it follows immediately that, for an arbitrary given x Î Cⁿ, the functions

x(t) = e^tAx

is a solution of the homogeneous equation (3). In fact

dx(t) dt

=

d e^tAx dt

= Ae^tAx = Ax(t).

Thus, the homogeneous equation (3) has infinitely many solutions. Therefore, by Theorem 1, the inhomogeneous equation (2), if has a solution, also has infinitely many of them. Each solution can be represented as a parameterized "generalized" curve in the space Cⁿ (a standard curve if n = 2 or n = 3). The parameter is t, which usually can be interpreted as "time". In order to determine a solution uniquely, one need to specify an initial condition, requiring that the curve passes through a given point at a given time t₀, i.e.

x1(t0) = x1^(0),  x2(t0) = x2^(0),  ..., xn(t0).

(5)

Conditions (5) can be written in the vector form

x(t0) = x0.

(6)

where x₀ is the vector with components x⁽⁰⁾_i, i = 1,2,...,n. The problem of solving equation (2) subject to the initial condition (6) is called the initial value problem, or the Cauchy problem. Usually it is written in the following form:

ì í î

x¢(t) = Ax(t), x(t0) = x0.

(7)

Assuming that the function f(t) is continuous (i.e. f_i(t) are continuous for all i = 1,2,...), one can use results of chapter 7 on the exponential functions e^tA to solve equation (2) in the following way:

Apply to the both parts of equation (2) the matrix e^-tA, we have

e^-tAx¢(t) = e^-tAAx(t)+e^-tAf(t).

(8)

From (8) we have

e^-tAx¢(t)-e^-tAAx(t) = e^-tAf(t).

(9)

By Theorem ? of chapter 7, equation (9) can be written as

d (e^-tAx(t)) dt

= e^-tAf(t).

Therefore,

e^-tAx(t) =

ó õ

t t0

e^-sAf(s)ds+c,

(10)

where  c  is an arbitrary constant vector. The constant vector  c is determined by the initial condition (6). Indeed, equation (10) with t = t₀ implies

e^-t0^Ax(t0) = e^t0^Ax0 =

ó õ

t0 t0

e^-sAf(s)ds+c = c,

hence (10) becomes

e^-tAx(t) = e^-t0^Ax0+

ó õ

t t0

e^-sAf(s)ds.

(11)

Finally, applying e^tA to both parts of (11), we obtain

x(t) = e^tAe^-t0^Ax0+e^tA

ó õ

t t0

e^-sAf(s)ds =

e^(t-t0^)Ax0+

ó õ

t t0

e^(t-s)Af(s)ds,

(12)

which is a solution to the initial value problem (7). Formula (12) is called the variation of parameters formula. It plays an important role in the theory of differential equations.

Example 1 Solve the initial value problem:

ì
í
î

x¢(t)	= x(t)+y(t)
y¢(t) = 3x(t)-y(t)

(13)

x(0) = -1,  y(0) = 1.

(14)

Solution: The above system can be written in the form

ì
í
î

x¢(t)	= Ax(t)
x(0)	= x0,

(15)

where

A =

é
ê
ë

1	1
3	-1

ù
ú
û

, x(t) =

é
ê
ë

x(t)

y(t)

ù
ú
û

,  x₀ =

é
ê
ë

-1

1

ù
ú
û

.

We need to find e^tA. The matrix tA has two eigenvalues l₁ = 2t, l₂ = -2t. Put f(l) = e^l, and we find the function r(l) = al+b, such that f(l₁) = f(l₁), r(l₂) = f(l₂). Hence, a and b must satisfy the following relations:

ì
í
î

2a+b	= e^2t
-2a+b = e^-2t,

from which it follows that a = [1/ 4t](e^2t-e^-2t), b = ¹/₂(e^2t+e^-2t). Hence

e^tA =

é
ê
ê
ê
ê
ê
ë

1 4	(3e^2t-e^-2t)

1 4	(e^2t-e^-2t)

1 4	(3e^2t-3e^-2t)

1 4	(e^2t+3e^-2t)

ù
ú
ú
ú
ú
ú
û

.

Thefore, the solution to the initial value problem (?) is

x(t) = e^tAx₀ =

é
ê
ê
ê
ê
ê
ë

1 2	(-e^-2t-e^2t)

1 2	(3e^-2t-e^2t)

ù
ú
ú
ú
ú
ú
û

.

(16)

From (?) we obtain the solutions x(t) and y(t) of the system (?).

x(t) =

1 2

(-e^-2t-e^2t),  y(t) =

1 2

(3e^-2t-e^2t).

Example 2 Solve the initial value problem:

ì
í
î

x¢(t)	= 3x(t)-y(t)+e^t
y¢(t) = 4x(t)-2y(t)

(17)

x(0) = 1,  y(0) = -1.

(18)

Solution:

8.2 Reduction of equations of higher orders.

The matrix method can also be used for solving linear differential equations with constant coefficients of arbitrary order n, as well as systems of such equations, by reducing them to systems of first order equations. Let us first examine in details the case of equations of second order.

A linear differential equation of second order with constant coefficients has the form

a2x¢¢(t)+a1x¢(t)+a0x(t) = g(t),

(19)

where a₂, a₁, a₀ are some scalars, g(t) is a known function. By a solution to this equation we meen any twice differentiable function x(t) such that when substituted in to (13), the equation is satisfied.

We may assume that a₂ ¹ 0 because otherwise we would have an equation of first order. By dividing by a₂, and denoting a = a₁/a₂, b = a₀/a₂,f(t) = g(t)/a₂, equation (13) will take the following form

x¢¢(t)+ax¢(t)+bx(t) = f(t).

(20)

The method of reduction consists in the following: define two new variables u₁ and u₂ by:

u₁(t) = x(t),  u₂(t) = x¢(t).

Then, as easily seen, x(t) is a solution of equation (14) if and only if (u₁, u₂) is a solution of the following system:

ì
í
î

u1¢(t)	= u2(t)
u2¢(t)	= -bu1(t)-au2(t)+f(t)

(21)

Thus, if we introduce the matrix A and vector functions u(t),  f(t) by

A =

é
ê
ë

0	1
-b	-a

ù
ú
û

,  u(t) =

é
ê
ë

u₁(t)

u₂(t)

ù
ú
û

,  f(t) =

é
ê
ë

0

f(t)

ù
ú
û

,

then system (15) will take form

u¢(t) = Au(t)+f(t),

(22)

In order for the considered equation to have a unique solution, we must specify two initial conditions

x(t₀) = x₀, x¢(t₀) = x₁,

which is equivalent to the condition

u(t0) = u0 =

é ê ë

x0 x1

ù ú û

.