Unit 2 - Notes

MTH166

Unit 2: Differential equations of higher order

1. Introduction to Linear Differential Equations

A Linear Differential Equation (LDE) of order $n$ is a differential equation in which the dependent variable ( $y$ ) and all its derivatives ( $y', y'', \dots, y^{(n)}$ ) occur only in the first degree and are not multiplied together.

1.1 General Form

The general form of an $n$ -th order linear differential equation is:

a_n(x)\frac{d^ny}{dx^n} + a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}} + \dots + a_1(x)\frac{dy}{dx} + a_0(x)y = Q(x)

Where:

$a_n(x), a_{n-1}(x), \dots, a_0(x)$ are coefficients (functions of $x$ or constants).
$a_n(x) \neq 0$ .
$Q(x)$ is a function of $x$ only.

1.2 Classification

Homogeneous LDE: If $Q(x) = 0$ for all $x$ in the interval.
$a_n(x)y^{(n)} + \dots + a_0(x)y = 0$
Non-Homogeneous LDE: If $Q(x) \neq 0$ .

1.3 Constant Coefficients

If the coefficients $a_n, \dots, a_0$ are real constants, the equation is called a Linear Differential Equation with Constant Coefficients. This unit focuses primarily on this type.

2. Solution of Linear Differential Equation: Structure

The complete solution (general solution) of a linear differential equation consists of two parts depending on whether it is homogeneous or non-homogeneous.

2.1 The General Solution Structure

For a non-homogeneous equation:

y(x) = y_c(x) + y_p(x)

$y_c(x)$ (Complementary Function - C.F.): This is the general solution to the associated homogeneous equation (setting $Q(x)=0$ ). It contains $n$ arbitrary constants.
$y_p(x)$ (Particular Integral - P.I.): This is a specific solution that satisfies the non-homogeneous equation (yielding $Q(x)$ ). It contains no arbitrary constants.

Note: For Homogeneous equations ( $Q(x)=0$ ), the solution is simply $y(x) = y_c(x)$ .

3. Linear Dependence and Linear Independence of Solutions

To form the Complementary Function ( $y_c$ ), we must find $n$ linearly independent solutions.

3.1 Linearly Dependent Functions

A set of functions $y_1, y_2, \dots, y_n$ is linearly dependent on an interval $I$ if there exist constants $c_1, c_2, \dots, c_n$ , not all zero, such that:

c_1y_1(x) + c_2y_2(x) + \dots + c_ny_n(x) = 0

for all

x

I

. (Essentially, one function can be written as a combination of the others).

3.2 Linearly Independent Functions

The functions are linearly independent if the equation above holds only when $c_1 = c_2 = \dots = c_n = 0$ .

3.3 The Wronskian ( $W$ )

The Wronskian determinant is a tool used to test linear independence. For $n$ functions $y_1, \dots, y_n$ :

W(y_1, \dots, y_n) = \begin{vmatrix} y_1 & y_2 & \dots & y_n \\ y_1' & y_2' & \dots & y_n' \\ \vdots & \vdots & \ddots & \vdots \\ y_1^{(n-1)} & y_2^{(n-1)} & \dots & y_n^{(n-1)} \end{vmatrix}

If $W \neq 0$ for at least one point in the interval, the functions are Linearly Independent.
If $W = 0$ for all $x$ , the functions are Linearly Dependent.

3.4 Fundamental Set of Solutions

If $y_1, y_2, \dots, y_n$ are $n$ linearly independent solutions of an $n$ -th order homogeneous linear differential equation, they form a Fundamental Set of Solutions. The general solution is:

y = c_1y_1 + c_2y_2 + \dots + c_ny_n

4. Method of Solution: The Differential Operator ( $D$ )

The symbol $D$ is used to represent the operation of differentiation with respect to the independent variable (usually $x$ ).

$D = \frac{d}{dx}$
$D^2 = \frac{d^2}{dx^2}$
$D^n = \frac{d^n}{dx^n}$

4.1 Algebraic Properties of $D$

The operator $D$ behaves algebraically like a polynomial variable when coefficients are constant:

Polynomial Form: An $n$ -th order LDE can be written as $f(D)y = Q(x)$ , where:
$f(D) = a_n D^n + a_{n-1} D^{n-1} + \dots + a_1 D + a_0$
Commutative Law: $(D-a)(D-b)y = (D-b)(D-a)y$ .
Operation on Exponentials: $f(D)e^{ax} = f(a)e^{ax}$ .

5. Solution of Second Order Homogeneous Linear DE with Constant Coefficients

Consider the equation:

a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = 0

In operator notation:

(aD^2 + bD + c)y = 0

5.1 The Auxiliary Equation (A.E.)

To solve this, we assume a solution of the form $y = e^{mx}$ . Substituting this into the DE yields the Auxiliary Equation (also called Characteristic Equation):

am^2 + bm + c = 0

We solve this quadratic equation for $m$ . Let the roots be $m_1$ and $m_2$ . The nature of the solution depends on the discriminant $\Delta = b^2 - 4ac$ .

5.2 Case 1: Real and Distinct Roots ( $\Delta > 0$ )

If $m_1$ and $m_2$ are real numbers and $m_1 \neq m_2$ .
The linearly independent solutions are $e^{m_1x}$ and $e^{m_2x}$ .

General Solution:

y = c_1 e^{m_1x} + c_2 e^{m_2x}

5.3 Case 2: Real and Repeated Roots ( $\Delta = 0$ )

If $m_1 = m_2 = m$ (a real root repeated twice).
One solution is $e^{mx}$ . To find the second independent solution, we multiply by $x$ .

General Solution:

y = (c_1 + c_2x)e^{mx}

5.4 Case 3: Complex Conjugate Roots ( $\Delta < 0$ )

If the roots are complex, they occur in conjugate pairs: $m_1 = \alpha + i\beta$ and $m_2 = \alpha - i\beta$ (where $i = \sqrt{-1}$ ).
The solution involves Euler's formula ( $e^{i\theta} = \cos\theta + i\sin\theta$ ).

General Solution:

y = e^{\alpha x}(c_1 \cos \beta x + c_2 \sin \beta x)

6. Solution of Higher Order Homogeneous Linear DE with Constant Coefficients

The methods for second-order equations extend directly to $n$ -th order equations.
Given: $a_n D^n y + \dots + a_0 y = 0$ .
Auxiliary Equation: $a_n m^n + \dots + a_0 = 0$ .

This polynomial of degree $n$ will have $n$ roots.

6.1 Rules for Constructing the General Solution

Nature of Roots	Roots ( $m$ )	Component of General Solution ( $y_c$ )
Distinct Real	$m_1, m_2, \dots, m_k$	$c_1 e^{m_1x} + c_2 e^{m_2x} + \dots + c_k e^{m_k x}$
Repeated Real	$m$ (repeated $k$ times)	$(c_1 + c_2x + c_3x^2 + \dots + c_k x^{k-1})e^{mx}$
Distinct Complex	$\alpha \pm i\beta$	$e^{\alpha x}(c_1 \cos \beta x + c_2 \sin \beta x)$
Repeated Complex	$\alpha \pm i\beta$ (repeated $k$ times)	$e^{\alpha x}[(c_1 + c_2x + \dots)\cos \beta x + (c_{k+1} + c_{k+2}x + \dots)\sin \beta x]$

6.2 Examples of Higher Order Solutions

Example A: Third Order (Distinct Roots)

Equation: $(D^3 - 6D^2 + 11D - 6)y = 0$
A.E.: $m^3 - 6m^2 + 11m - 6 = 0$
Roots: Factoring yields $(m-1)(m-2)(m-3)=0 \implies m = 1, 2, 3$ .
Solution:

y = c_1 e^{x} + c_2 e^{2x} + c_3 e^{3x}

Example B: Fourth Order (Repeated Roots)

Equation: $(D^4 - 2D^3 + D^2)y = 0$
A.E.: $m^2(m^2 - 2m + 1) = m^2(m-1)^2 = 0$
Roots: $m = 0, 0, 1, 1$ . (0 is repeated twice, 1 is repeated twice).
Solution:

y = (c_1 + c_2x)e^{0x} + (c_3 + c_4x)e^{1x}

y = c_1 + c_2x + (c_3 + c_4x)e^{x}

Example C: Fourth Order (Repeated Complex Roots)

Equation: $(D^2 + 1)^2 y = 0$
A.E.: $(m^2 + 1)^2 = 0 \implies (m^2+1)(m^2+1)=0$
Roots: $m = \pm i, \pm i$ (repeated twice). Here $\alpha = 0, \beta = 1$ .
Solution:

y = e^{0x}[(c_1 + c_2x)\cos(1x) + (c_3 + c_4x)\sin(1x)]

y = (c_1 + c_2x)\cos x + (c_3 + c_4x)\sin x

7. Summary Checklist for Solving

Identify the Order: Note the highest derivative $n$ .
Verify Linearity and Coefficients: Ensure coefficients are constants and the equation is linear.
Check Homogeneity: Is the right side 0?
- If Yes: You are solving for the final solution $y$ .
- If No: You are solving only for $y_c$ (Complementary Function).
Write Operator Form: Replace $d^k/dx^k$ with $D^k$ .
Form Auxiliary Equation: Replace $D$ with $m$ and set to 0.
Calculate Roots: Find all $n$ roots (real, complex, distinct, or repeated).
Apply Rules: Construct the linear combination based on the nature of the roots.

Unit 2 - Notes

Table of Contents

Unit 2: Differential equations of higher order

1. Introduction to Linear Differential Equations

1.1 General Form

1.2 Classification

1.3 Constant Coefficients

2. Solution of Linear Differential Equation: Structure

2.1 The General Solution Structure

3. Linear Dependence and Linear Independence of Solutions

3.1 Linearly Dependent Functions

3.2 Linearly Independent Functions

3.3 The Wronskian ()

3.4 Fundamental Set of Solutions

4. Method of Solution: The Differential Operator ()

4.1 Algebraic Properties of

5. Solution of Second Order Homogeneous Linear DE with Constant Coefficients

5.1 The Auxiliary Equation (A.E.)

5.2 Case 1: Real and Distinct Roots ()

5.3 Case 2: Real and Repeated Roots ()

5.4 Case 3: Complex Conjugate Roots ()

6. Solution of Higher Order Homogeneous Linear DE with Constant Coefficients

6.1 Rules for Constructing the General Solution

6.2 Examples of Higher Order Solutions

Example A: Third Order (Distinct Roots)

Example B: Fourth Order (Repeated Roots)

Example C: Fourth Order (Repeated Complex Roots)

7. Summary Checklist for Solving

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3.3 The Wronskian ( $W$ )

4. Method of Solution: The Differential Operator ( $D$ )

4.1 Algebraic Properties of $D$

5.2 Case 1: Real and Distinct Roots ( $\Delta > 0$ )

5.3 Case 2: Real and Repeated Roots ( $\Delta = 0$ )

5.4 Case 3: Complex Conjugate Roots ( $\Delta < 0$ )