INTRODUCTION

In this section we are going to consider an alternative method for solving a homogeneous system of linear first-order differential equations. This method is applicable to such a system X′ = AX whenever the coefficient matrix A is diagonalizable.

Coupled Systems

A homogeneous linear system X′ = AX,

(1)

in which each is expressed as a linear combination of x₁, x₂, … , x_n is said to be coupled. If the coefficient matrix A is diagonalizable, then the system can be uncoupled in that each can be expressed solely in terms of x_i.

If the matrix A has n linearly independent eigenvectors then we know from Theorem 8.12.2 that we can find a matrix P such that P⁻¹AP = D, where D is a diagonal matrix. If we make the substitution X = PY in the system X′ = AX, then

PY′ = APY or Y′ = P⁻¹APY or Y′ = DY.(2)

The last equation in (2) is the same as

(3)

Since D is diagonal, an inspection of (3) reveals that this new system is uncoupled: Each differential equation in the system is of the form = λ_i y_i , i = 1, 2, … , n. The solution of each of these linear equations is y_i = c_i, i = 1, 2, … , n. Hence the general solution of (3) can be written as the column vector

(4)

Since we now know Y and since the matrix P can be constructed from the eigenvectors of A, the general solution of the original system X′ = AX is obtained from X = PY.

EXAMPLE 1 Uncoupling a Linear System

Solve X′ = X by diagonalization.

SOLUTION

We begin by finding the eigenvalues and corresponding eigenvectors of the coefficient matrix.

From det(A − λI) = −(λ + 2)(λ − 1)(λ − 5), we get λ₁ = −2, λ₂ = 1, and λ₃ = 5. Since the eigenvalues are distinct, the eigenvectors are linearly independent. Solving (A − λ_iI)K = 0 for i = 1, 2, and 3 gives, respectively,

(5)

Thus a matrix that diagonalizes the coefficient matrix is

The entries on the main diagonal of D are the eigenvalues of A corresponding to the order in which the eigenvectors appear in P:

As we have shown previously, the substitution X = PY in X′ = AX gives the uncoupled system Y′ = DY. The general solution of this last system is immediate:

Hence the solution of the given system is

(6)≡

Note that (6) can be written in the usual manner by expressing the last matrix as a sum of column matrices:

Solution by diagonalization will always work provided we can find n linearly independent eigenvectors of the n × n matrix A; the eigenvalues of A could be real and distinct, complex, or repeated. The method fails when A has repeated eigenvalues and n linearly independent eigenvectors cannot be found. Of course, in this last situation A is not diagonalizable.

Since we have to find eigenvalues and eigenvectors of A, this method is essentially equivalent to the procedure presented in the last section.

In the next section we shall see that diagonalization can also be used to solve nonhomogeneous linear systems X′ = AX + F(t).

10.3 Exercises Answers to selected odd-numbered problems begin on page ANS-28.

In Problems 1–10, use diagonalization to solve the given system.

1. 
2. 
3. 
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10. 
11. We have already shown how to solve the system of linear second-order differential equations that describes the motion of the coupled spring/mass system in Figure 3.12.1,

    (7)

    in three different ways (see Example 4 in Section 3.12, Problem 56 in Exercises 10.2, and Example 1 in Section 4.6). In this problem you are led through the steps of how (7) can also be solved using diagonalization.

    1. Express (7) in the form MX″ + KX = 0, where X = . Identify the 2 × 2 matrices M and K. Explain why the matrix M has an inverse.
    2. Express the system in part (a) as

       X″ + BX = 0.(8)

       Identify the matrix B.

    3. Solve system (7) in the special case where m₁ = 1, m₂ = 1, k₁ = 3, and k₂ = 2 by solving (8) using the diagonalization method. In other words, let X = PY, where P is a matrix whose columns are the eigenvectors of B.
    4. Show that your solution X in part (c) is the same as that given in part (d) of Problem 56 in Exercises 10.2.