System of First Order Differential Equations

System of First-order Linear Differential Equations




CHAPTER 3
SYSTEM OF FIRST-ORDER LINEAR
DIFFERENTIAL EQUATIONS
Concept Mapping
System of first-order linear
differential equations

Formation and solution



Homogeneous Nonhomogeneous
' AY Y= AY+G

Matrix A has:
Case 1: Real and different Method of Variation of
eigenvalues undetermined parameter
Case 2: Same or repeated coefficient
eigenvalues For any G
For G
Case 3: Complex
number
eigenvalues exponent,
=atBi polynomial,
trigonometry

Modelling of RLC circuit

Objective
At the end ofthis chapter, students should be able to:
(a) solve homogeneous system of first-order linear differential equations
through finding eigenvalues and its corresponding eigenvectors.
(b) solve non homogeneous system of first-order linear differential equations
by using
(1) method of undetermined coefficient.
(ii) variation of parameters.
(c) Model elementary RLC circuit into system of first-order linear
differential equations and solve it.


Key Term (English - Bahasa Melayu)

corresponding eigenvector eigen vektor yang sepadan
eigenvalue nilai eigen
linearly independent bersandar secara linear
matrices matriks
system of first-order linear sistem persamaan pembezaan linear
differential equations peringkat pertama

, System of First-order Lincar Differential Equations



solve system of first-
In chapter, will discuss the analytical method to
this we

order linear differential equations.


Definition
System of First-order Linear Differential Equations
n a more general form, the system of first-order ordinary differential

equations is given as below.


(1)


We call () a linear system if it is linear in yi,y2, -. y,, that is, if it can be

written as
1+a12 t + any, +8,()
- (2)
=az1 +az22 ++azy, +82(X)

,1+,2) +.+ an,+8,(x)

In vector fom, (2) can be written as
Y= AY +G (3)
az1
a21 a22 2 2
where A =| Y =| G=82




nl 2 an
The system is called homogeneous if G=0, so that it is Y'=AY; If G#0,
then (3) is called non-homogeneous. For example, the system
=3+3y»»

is homogeneous; the system
-3+3y, + 8,
V +5y, +4e*,
is non-homogeneous.

We start thediscussion from the solution of system of
first-order linear
differential equations.

Example 3.1... . ***.
Given that a system consists of two first-order linear differential equations.
=3 +3y + 8,
+5y, +4e*
Show that y(x) 3e + - 4 -
=
and y2(x)= -e" +e" + is the
solution of system.

, System of Fist-order Linear Differential Equations




Solution:
Step 1: From )=3e + e - 4 - 9 , hence ) =6e + 6e -12e*.
From y,(x)=-e+e +3, hence y(x) = -2e + 6e*.
Step 2: From y =3y, +3y, +8
313e" +*-4e*-9)+3(-e" +e+})+8
= 9e +3e"-12e-10-3e2 +3e +2+8
=
6e +6e" 12e" -
=
y(x)
From yy +5y, +4e*
-3e+-4e4+s-e+e+)+4*
=3e+*-4e 1-Se +5e ++4e
-2e +6e" =y)
a ) and ,(a) is the solution ofsystem.
T r y Question 1, Exercise 3A



Exercise 3A

1. Show that y ) and y,(x) is the solution of system of first-order linear
differential equations below.
(a) +% ) = e* +e
4y+2 2(x)=2e3 -2e
(b) +4e" r ) = e* +4xe
=- x)=3e" -4xe



3.1 Homogeneous System of First-Order Linear
Differential Equations
In this section, we will learn to solve system of first-order linear
differential equations. We write the system into matrix form, then find the
eigenvalues and its corresponding eigenvectors. Similar to homogeneous
second-order differential equations in Chapter 2, there are 3 cases for
homogeneous system, namely

Case 1: Real and different eigenvalues, A, *, *...,
Case 2: Same or repeated eigenvalues, 4 = /2 . . . = 4,
Case 3: Complex number eigenvalues, 2 =atfi

We start our discussion-with the homogeneous system consists of two
first-order lincar differential equations.