Second Otder Ordinry Differential Equations
eHAPTER 2
o R D rN A RY BEr FcFoEH Sfii I ? e uAr r o Ns
Concept Mapping
Second order ordinary differential equations
(constant coefficients and linear)
Homogeneous Non-homogeneous
ay' (x) + by' (x) + cy(x) = Q ay' (x) + by' (x) + cy(x) = f (x)
am'+bm+c=0
Case l: m, * m. andreal Method of Variation of
y(x) = Ae*r* + Be*2x or undetermined parameter
y(x) = A cosh mrx + B sinh mzx coefficient
Case2: ffi=ffi:=mz andreal I
y(x) =(A+ Bx)e*' For /(x) For any continuous
Case3: ry=s*Bi I Exponent function of f (x)
y(x) = ee (Acos px + B sin px) il Polynomial
III Trigonometry
(sin and cos only)
IV Combination or
multiplication of I, II
and III
Modelling RZC circuit
Objective
At the end of this chapter, students should be able to
(a) solve second order linear differential eiiuations with constant coefficient
by using
(i) method of undetermined coefficient,
(iD variation of parameter.
(b) model elementary RLC circuit into second order linear differential
equations with constant coefficients and solve it.
Key Term (English - Bahasa Melayu)
Constant coefficient Pekali malar
Homogeneous equation Persamaan homogen
Non-homogeneous equation Persamaan tak homogen
Second order ordinary differential Persamaan pembezaan biasa peringkat
equations kedua
Undetermined coefficient Pekali tak tentu
Variation of parameter Ubahan parameter
41
, Second Order Qdinary Differential Equations
In this chapter, we will tliscuss the methods to solve second order
linear differential equations with constant coefficients. First of all, we need to
understand terms such as homogertous equation, non-homogeneous equation,
and constant coefficient.
Say, for the second order iifferential equations in the form of
ay' (x) + by' (x) + cy(x) = -f (x),
(a) notethatthepowerfor y'(x), y'(x) and y(x) are l.Thesecondorder
differential equations in this form are called second order linear
differential equations.
(b) if a, b and c are constants, then the differential equation is with
constant coefficients.
(c) if a, b and c are constants and f (x) = 0, then we obtain second order
hom o geneous linear differential equations with constant coeffi cients.
(d) if a, b and c are constants and "f (x) * 0, then we obtain second order
'non-homogeneous
linear differential equations with constant
coeffrcients.
Example 2.1
Determine whether the following second order differential equation is linear or
not linear, with constant coefficients or
with variable coefficients,
homogeneous or non-homogeneous.
(a) d',+
clx'
= -a, (d) 3t4 = -4x
dx'
(b) yn-4y'tx=0 (e) y' - 4y' * xy =0
(c) (y'Y-4y'+x=0
Solution:
Rewrite in the form of ay'(x)+by'(x)+cy(x)=.f(x), if a, b and c are
constants, then the differential equation is with constant coefficients.
Linear? Homogeneous? Constant
coefficient?
(a) !'+4Y=g
(b) yo-4y'=-x x
(c) ,c *"
0'Y -4y'=-*
(d) 3y" = -4x ,c
(e) Yo-4Y'-lxy=0 x
will
; ;;;;;; ;;;";, ;;;ffi;'::;;:
use the series solution techniques to solve it. However, in this chapter, we
deal with second order linear differential equations with constant cofficients
only.
42
, Second Order Ordinary Differential Equations
Exercise 2A
1. Chek whether the following differential equations is second order, linear,
horflogeneous and with constant coefficients' or not.
Order Linear Homogeneous Coefficient
(a) y" -4x2 =o
(b) y'(x)-10y2(x; = 6
(c) y'+(sinx)y'-2=0
(d) y"=y
Answers to Exercise 2A
Order Linear Homoqeneous Coefficient
(a) Second order Linear Non-homoseneous Constant
(b) Second order Not linear Homogeneous Constant
(c) Second order Linear Non-homogeneous Variable
(d) Second order Linear Homogeneous Constant
In this section, we will solve the second order homogeneous linear
differential equation with constant coefficients. We can use calculator to help
us to solve quadratic equations, as follows.
Step l: Set the mode of calculator to mode Equation tnq$.
@@@@
Step2: Define the degree .
O@
Step 3: Define a, b andc in equatiorr a)c' + bx +c = 0 accordingly.
Step 4: Press @ to obtain the first root and pr.r, @ againto obtain the
second root.
If the calculator screen shows R<+r on the right top, press @ @ to obtain
the value for imaginary part.
Step l: Set the mode of calculator to mode Equation (EQN).
@@
Steo2: Define the degree .
@ axz+bx*c=0
Step 3 & 4: Same as above
43
eHAPTER 2
o R D rN A RY BEr FcFoEH Sfii I ? e uAr r o Ns
Concept Mapping
Second order ordinary differential equations
(constant coefficients and linear)
Homogeneous Non-homogeneous
ay' (x) + by' (x) + cy(x) = Q ay' (x) + by' (x) + cy(x) = f (x)
am'+bm+c=0
Case l: m, * m. andreal Method of Variation of
y(x) = Ae*r* + Be*2x or undetermined parameter
y(x) = A cosh mrx + B sinh mzx coefficient
Case2: ffi=ffi:=mz andreal I
y(x) =(A+ Bx)e*' For /(x) For any continuous
Case3: ry=s*Bi I Exponent function of f (x)
y(x) = ee (Acos px + B sin px) il Polynomial
III Trigonometry
(sin and cos only)
IV Combination or
multiplication of I, II
and III
Modelling RZC circuit
Objective
At the end of this chapter, students should be able to
(a) solve second order linear differential eiiuations with constant coefficient
by using
(i) method of undetermined coefficient,
(iD variation of parameter.
(b) model elementary RLC circuit into second order linear differential
equations with constant coefficients and solve it.
Key Term (English - Bahasa Melayu)
Constant coefficient Pekali malar
Homogeneous equation Persamaan homogen
Non-homogeneous equation Persamaan tak homogen
Second order ordinary differential Persamaan pembezaan biasa peringkat
equations kedua
Undetermined coefficient Pekali tak tentu
Variation of parameter Ubahan parameter
41
, Second Order Qdinary Differential Equations
In this chapter, we will tliscuss the methods to solve second order
linear differential equations with constant coefficients. First of all, we need to
understand terms such as homogertous equation, non-homogeneous equation,
and constant coefficient.
Say, for the second order iifferential equations in the form of
ay' (x) + by' (x) + cy(x) = -f (x),
(a) notethatthepowerfor y'(x), y'(x) and y(x) are l.Thesecondorder
differential equations in this form are called second order linear
differential equations.
(b) if a, b and c are constants, then the differential equation is with
constant coefficients.
(c) if a, b and c are constants and f (x) = 0, then we obtain second order
hom o geneous linear differential equations with constant coeffi cients.
(d) if a, b and c are constants and "f (x) * 0, then we obtain second order
'non-homogeneous
linear differential equations with constant
coeffrcients.
Example 2.1
Determine whether the following second order differential equation is linear or
not linear, with constant coefficients or
with variable coefficients,
homogeneous or non-homogeneous.
(a) d',+
clx'
= -a, (d) 3t4 = -4x
dx'
(b) yn-4y'tx=0 (e) y' - 4y' * xy =0
(c) (y'Y-4y'+x=0
Solution:
Rewrite in the form of ay'(x)+by'(x)+cy(x)=.f(x), if a, b and c are
constants, then the differential equation is with constant coefficients.
Linear? Homogeneous? Constant
coefficient?
(a) !'+4Y=g
(b) yo-4y'=-x x
(c) ,c *"
0'Y -4y'=-*
(d) 3y" = -4x ,c
(e) Yo-4Y'-lxy=0 x
will
; ;;;;;; ;;;";, ;;;ffi;'::;;:
use the series solution techniques to solve it. However, in this chapter, we
deal with second order linear differential equations with constant cofficients
only.
42
, Second Order Ordinary Differential Equations
Exercise 2A
1. Chek whether the following differential equations is second order, linear,
horflogeneous and with constant coefficients' or not.
Order Linear Homogeneous Coefficient
(a) y" -4x2 =o
(b) y'(x)-10y2(x; = 6
(c) y'+(sinx)y'-2=0
(d) y"=y
Answers to Exercise 2A
Order Linear Homoqeneous Coefficient
(a) Second order Linear Non-homoseneous Constant
(b) Second order Not linear Homogeneous Constant
(c) Second order Linear Non-homogeneous Variable
(d) Second order Linear Homogeneous Constant
In this section, we will solve the second order homogeneous linear
differential equation with constant coefficients. We can use calculator to help
us to solve quadratic equations, as follows.
Step l: Set the mode of calculator to mode Equation tnq$.
@@@@
Step2: Define the degree .
O@
Step 3: Define a, b andc in equatiorr a)c' + bx +c = 0 accordingly.
Step 4: Press @ to obtain the first root and pr.r, @ againto obtain the
second root.
If the calculator screen shows R<+r on the right top, press @ @ to obtain
the value for imaginary part.
Step l: Set the mode of calculator to mode Equation (EQN).
@@
Steo2: Define the degree .
@ axz+bx*c=0
Step 3 & 4: Same as above
43
