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Mathematics LibreTexts

10.2: Linear Systems of Differential Equations

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A first order system of differential equations that can be written in the form

y1=a11(t)y1+a12(t)y2++a1n(t)yn+f1(t)y2=a21(t)y1+a22(t)y2++a2n(t)yn+f2(t)yn=an1(t)y1+an2(t)y2++ann(t)yn+fn(t)

is called a linear system.

The linear system Equation ??? can be written in matrix form as

[y1y2yn]=[a11(t)12(t)1n(t)a21(t)22(t)2n(t)an1(t)n2(t)nn(t)][y1y2yn]+[f1(t)f2(t)fn(t)],

or more briefly as

y=A(t)y+f(t),

where

y=[y1y2yn],A(t)=[a11(t)12(t)1n(t)a21(t)22(t)2n(t)an1(t)n2(t)nn(t)],andf(t)=[f1(t)f2(t)fn(t)].

We call A the coefficient matrix of Equation ??? and f the forcing function. We’ll say that A and f are continuous if their entries are continuous. If f=0, then Equation ??? is homogeneous; otherwise, Equation ??? is nonhomogeneous.

An initial value problem for Equation ??? consists of finding a solution of Equation ??? that equals a given constant vector

k=[k1k2kn].

at some initial point t0. We write this initial value problem as

y=A(t)y+f(t),y(t0)=k.

The next theorem gives sufficient conditions for the existence of solutions of initial value problems for Equation ???. We omit the proof.

Theorem 10.2.1 : Existence

Suppose the coefficient matrix A and the forcing function f are continuous on (a,b), let t0 be in (a,b), and let k be an arbitrary constant n-vector. Then the initial value problem

y=A(t)y+f(t),y(t0)=k

has a unique solution on (a,b).

Example 10.2.1

  1. Write the system y1=2y1+2y2+2e4ty2=2y1+2y2+2e4t in matrix form and conclude from Theorem 10.2.1 that every initial value problem for Equation ??? has a unique solution on (,).
  2. Verify that y=15[87]e4t+c1[11]e3t+c2[11]et is a solution of Equation ??? for all values of the constants c1 and c2.
  3. Find the solution of the initial value problem y=[1221]y+[21]e4t,y(0)=15[322].
Solution a

The system Equation ??? can be written in matrix form as

y=[12]y+[21]e4t.

An initial value problem for Equation ??? can be written as

y=[1221]y+[21]e4t,y(t0)=[k1k2].

Since the coefficient matrix and the forcing function are both continuous on (,), Theorem 10.2.1 implies that this problem has a unique solution on (,).

Solution b

If y is given by Equation ???, then

Ay+f=15[1221][87]e4t+c1[1221][11]e3t+c2[1221][11]et+[21]e4t=15[2223]e4t+c1[33]e3t+c2[11]et+[21]e4t=15[3228]e4t+3c1[11]e3tc2[11]et=y.

Solution c

We must choose c1 and c2 in Equation ??? so that

15[87]+c1[11]+c2[11]=15[322],

which is equivalent to

[1111][c1c2]=[13].

Solving this system yields c1=1, c2=2, so

y=15[87]e4t+[11]e3t2[11]et

is the solution of Equation ???.

Note

The theory of n×n linear systems of differential equations is analogous to the theory of the scalar n-th order equation P0(t)y(n)+P1(t)y(n1)++Pn(t)y=F(t) as developed in Sections 9.1. For example by rewriting Equation ??? as an equivalent linear system it can be shown that Theorem 10.2.1 implies Theorem 9.1.1.


This page titled 10.2: Linear Systems of Differential Equations is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.

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