Linear Differential Equations of Higher Order: Concepts and Applications, Lecture notes of Differential Equations

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LINEAR DIFFERENTIAL

EQUATIONS OF HIGHER

ORDER

INTENDED LEARNING OUTCOMES:

• Identify Homogeneous and Non-homogeneous Linear D.E.’s;
• Understand the properties of the Linear D.E.’s of Higher Order and the Linear Combination of Functions;
• Identify the linear independence of functions;
• Understand the concept of Differential Operator; and
• Evaluate Linear D.E.’s of Higher Order in Operator Form.

SAMPLE PROBLEMS Determine the respective order, degree, unknown variable, independent variable, homogeneity, and linearity of the following D.E.’s:

1. 𝐲𝐲 ′′ + 𝐲 ′ + 𝐲 = 𝟏
2. 𝐝

𝐲 𝐝𝐱

𝐝

𝐲 𝐝𝐱

= 𝐬𝐢𝐧 𝐲

𝐝

𝐲 𝐝𝐱

𝐝𝐲 𝐝𝐱

• 𝐲 𝟐 = 𝟎

4. 𝐞 𝐱 𝐝

𝐲 𝐝𝐱

• 𝐱 𝐝𝐲 𝐝𝐱
• 𝐲 = 𝟐

5. 𝐱 𝟑 𝐝

𝐲 𝐝𝐱

• 𝐱 𝟐 𝐝𝐲 𝐝𝐱
• 𝐱 𝟐 𝐲 = 𝐬𝐢𝐧 𝐱

nd

order, 1

st

degree, y, x, Non-Linear

th

order, 1

st

degree, y, x, Non-Linear

rd

order, 1

st

degree, y, x, Non-Linear

nd

order, 1

st

degree, y, x, Linear, variable

coefficient, non-homogeneous

nd

order, 1

st

degree, y, x, Linear, variable

coefficient, non-homogeneous

SAMPLE PROBLEMS Re-write the given D.E.’s and determine their respective order, degree, unknown variable, independent variable, linearity, homogeneity,

6. 𝐲 ′′′ + 𝐲 ′′ + 𝐲 ′ + 𝐲 = 𝟏
7. 𝐝

𝐮 𝐝𝐭

• 𝐭 𝐝𝐮 𝐝𝐭
• 𝟑𝐮 = 𝟏

8. 𝐲 ′ = − 𝐱 𝐲 ; 𝐲𝐲 ′ = −𝐱
9. 𝐝

𝐲 𝐝𝐱

𝐝𝐲 𝐝𝐱 = 𝐱 𝟐 𝐲

𝐝

𝐲 𝐝𝐱

𝐝𝐲 𝐝𝐱

• 𝐲 = 𝟎

rd

order, 1

st

degree, y, x, Linear, constant

coefficient, non-homogeneous

nd

order, 1

st

degree, u, t, Linear, variable

coefficient, non-homogeneous

st

order, 1

st

degree, y, x, Non-Linear

nd

order, 1

st

degree, y, x, Linear , variable

coefficient, non-homogeneous

rd

order, 1

st

degree, y, x, Linear, constant

coefficient, homogenous

FIRST–ORDERED LINEAR DIFFERENTIAL EQUATIONS

Solution:

׬ 𝑷𝒅𝒙^

FIRST–ORDERED LINEAR DIFFERENTIAL EQUATIONS

Solution:

׬ 𝑷𝒅𝒙^

FIRST–ORDERED LINEAR DIFFERENTIAL EQUATIONS

Solution:

HIGHER–ORDERED LINEAR DIFFERENTIAL EQUATIONS

If y = Yp is any particular solution of a general linear differential equation and y = Yc

is the general solution of its corresponding homogeneous equation, then y = Yp + Yc

𝑛 𝑦 𝑑𝑥𝑛^

𝑛− 1 𝑦 𝑑𝑥𝑛−^1

𝑛 (Yp + Yc) 𝑑𝑥 𝑛

𝑛− 1 (Yp + Yc) 𝑑𝑥 𝑛− 1

𝑑(Yp + Yc) 𝑑𝑥

• 𝑎𝑛 (Yp + Yc) = 𝑓(𝑥) 𝑎 0

𝑛 (Yp) 𝑑𝑥 𝑛

𝑛 (Yc) 𝑑𝑥 𝑛

• … + 𝑎𝑛 Yp + 𝑎𝑛 Yc = 𝑓(𝑥) 𝑎 0

𝑛 (Yp) 𝑑𝑥𝑛^

• ⋯ + 𝑎𝑛 Yp + 𝑎 0

𝑛 (Yc) 𝑑𝑥𝑛^

• … + 𝑎𝑛 Yc = 𝑓(𝑥) 𝑓(𝑥) = 𝑓(𝑥)

LINEAR COMBINATION OF FUNCTIONS

“Any linear combination of solutions of a linear differential equation is also a

solution.”

• If 𝑦 1 and 𝑦 2 are solutions to the homogeneous linear equation with constant coefficients, then

is a solution, where 𝑐 1 and 𝑐 2 are constants.

𝑛 𝑦 𝑑𝑥𝑛^

𝑛− 1 𝑦 𝑑𝑥𝑛−^1

Since 𝑦 1 and 𝑦 2 are solutions, then 𝑎 0 𝑑𝑛𝑦 1 𝑑𝑥𝑛^

𝑑𝑛−^1 𝑦 1 𝑑𝑥𝑛−^1

and 𝑎 0

𝑛 𝑦 2 𝑑𝑥𝑛^

𝑛− 1 𝑦 2 𝑑𝑥𝑛−^1

LINEAR COMBINATION OF FUNCTIONS 𝑎 0

𝑛 𝑦 𝑑𝑥 𝑛

𝑛− 1 𝑦 𝑑𝑥 𝑛− 1

Since 𝑦 1 and 𝑦 2 are solutions, then 𝑎 0 𝑑𝑛𝑦 1 𝑑𝑥𝑛^

𝑑𝑛−^1 𝑦 1 𝑑𝑥𝑛−^1

and 𝑎 0

𝑛 𝑦 2 𝑑𝑥 𝑛

𝑛− 1 𝑦 2 𝑑𝑥 𝑛− 1

Multiply the first equation above by 𝑐 1 and the second by 𝑐 2 then add and re-group the two resulting equations. 𝑎 0 𝑐 1

𝑛 𝑦 1 𝑑𝑥𝑛^

𝑛 𝑦 2 𝑑𝑥𝑛^

𝑛− 1 𝑦 1 𝑑𝑥𝑛−^1

𝑛− 1 𝑦 2 𝑑𝑥𝑛−^1

LINEAR INDEPENDENCE OF FUNCTIONS

“If

then the functions y

1

, y

2

, …, y

n

are said to be linearly independent, otherwise linearly

dependent. The determinant W is known as the Wronskian.”

LINEAR INDEPENDENCE OF FUNCTIONS

Example : Consider the functions 𝑦 1 = cos 2𝑥 , 𝑦 2 = sin 2𝑥 , and 𝑦 3 = cos 2𝑥 +

𝜋 3

and prove that

they are linearly dependent.

′ 𝑦 2 ′ … 𝑦𝑛 ′ 𝑦 1 ′′ 𝑦 2 ′′ … 𝑦𝑛 ′′ ⋮ ⋮ ⋮ ⋮ 𝑦 1 (𝑛− 1 ) 𝑦 2 (𝑛− 1 ) … 𝑦𝑛 (𝑛− 1 )

cos 2 𝑥 sin 2 𝑥 cos 2 𝑥 +

− 2 sin 2 𝑥 2 cos 2 x − 2 sin 2 𝑥 +

− 4 cos 2 𝑥 − 4 sin 2 𝑥 − 4 cos 2 𝑥 +

LINEAR INDEPENDENCE OF FUNCTIONS

Example : Show that the the functions 𝑦 1 = 1 , 𝑦 2 = 𝑥 , and 𝑦 3 = 𝑥

2

are linearly independent in all

intervals.

′ 𝑦 2 ′ … 𝑦𝑛 ′ 𝑦 1 ′′ 𝑦 2 ′′ … 𝑦𝑛 ′′ ⋮ ⋮ ⋮ ⋮ 𝑦 1 (𝑛− 1 ) 𝑦 2 (𝑛− 1 ) … 𝑦𝑛 (𝑛− 1 )

2 0 1 2 𝑥 0 0 2 𝑊 =

2 1 𝑥 0 1 2 𝑥 0 1 0 0 2 0 0 𝑊 = 2 + 0 + 0 − 0 + 0 + 0 𝑊 = 2 Hence, functions y1, y2, & y 3 are linearly independent.

DIFFERENTIAL OPERATOR, D

The concept of the differential operator D, a symbol used extensively in the solution of linear equations

of higher order, is here presented:

a. D denotes the operation of differentiation with respect to the independent variable, say x, that is 𝐷 =

𝑑 𝑑𝑥

EXAMPLES:

1. D 2 x 3 = 6 x 2
2. D x ln x = 1 + ln x
3. D e − 2 x = − 2 e − 2 x
4. D x 2 + cos 3 x − ln x = 2 x − 3 sin 3 x − 1 x

b. D

r

indicates the number of r times the function should be differentiated relative to the independent

variable.

EXAMPLES:

1. D 2 2 x 3 = 12 x
2. D 0 x ln x = x ln x
3. D 4 e − 2 x = 16 e − 2 x
4. D 3 x 2 + cos 3 x − ln x = 27 sin 3 x − 2 x^3