Unit 2 - Notes

MTH174 6 min read

Unit 2: Linear differential equation-I

1. Introduction to Linear Differential Equation

A Linear Differential Equation (LDE) is a differential equation in which the dependent variable (usually $y$ ) and all its derivatives appear only in the first degree, and they are not multiplied together.

The general form of an $n$ -th order linear differential equation is:
$a_n(x) \frac{d^n y}{dx^n} + a_{n-1}(x) \frac{d^{n-1} y}{dx^{n-1}} + \dots + a_1(x) \frac{dy}{dx} + a_0(x)y = f(x)$
Where:

$y$ is the dependent variable, and $x$ is the independent variable.
$a_0(x), a_1(x), \dots, a_n(x)$ are known functions of $x$ (or constants), called coefficients.
$a_n(x) \neq 0$ ensures the equation is of order $n$ .
$f(x)$ is a known function of $x$ .

Classification:

Homogeneous LDE: If $f(x) = 0$ for all $x$ , the equation is homogeneous.
Non-homogeneous LDE: If $f(x) \neq 0$ , the equation is non-homogeneous.
Constant Coefficients: If $a_n, a_{n-1}, \dots, a_0$ are constants rather than functions of $x$ .

2. Solution of Linear Differential Equation

The complete solution of a non-homogeneous linear differential equation consists of two parts:

Complementary Function (C.F. or $y_c$ ): This is the general solution of the corresponding homogeneous equation (setting $f(x) = 0$ ). It contains $n$ arbitrary constants for an $n$ -th order equation.
Particular Integral (P.I. or $y_p$ ): This is any specific solution to the non-homogeneous equation that contains no arbitrary constants.

The General Solution is given by:
$y(x) = y_c(x) + y_p(x)$
(Note: For homogeneous equations where $f(x)=0$ , the Particular Integral is zero, and the general solution is simply $y = y_c$ .)

3. Linear Dependence and Linear Independence of Solution

When finding the Complementary Function of an $n$ -th order LDE, we look for $n$ distinct solutions $y_1, y_2, \dots, y_n$ . These solutions must be linearly independent.

Linear Dependence

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 every $x$ in the interval.

Linear Independence

If the only way to satisfy the above equation is if all constants are zero ( $c_1 = c_2 = \dots = c_n = 0$ ), then the functions are linearly independent.

The Wronskian

To test for linear independence, we use the Wronskian determinant, defined as:
$W(y_1, y_2, \dots, y_n) = \begin{vmatrix} y_1 & y_2 & \dots & y_n \\ y_1' & y_2' & \dots & y_n' \\ \vdots & \vdots & \ddots & \vdots \\ y_1^{(n-1)} & y_2^{(n-1)} & \dots & y_n^{(n-1)} \end{vmatrix}$

If $W \neq 0$ for at least one point in the interval, the solutions are linearly independent.
If $W = 0$ for all $x$ in the interval, the solutions are linearly dependent.

4. Method of Solution of Linear Differential Equation - Differential Operator

To simplify the writing and solving of LDEs, we introduce the Differential Operator, denoted by $D$ .

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

Using this notation, an $n$ -th order LDE with constant coefficients can be written as:
$(a_n D^n + a_{n-1} D^{n-1} + \dots + a_1 D + a_0)y = f(x)$
Or simply:
$f(D)y = X$
where $f(D) = a_n D^n + a_{n-1} D^{n-1} + \dots + a_0$ is a polynomial in $D$ .

Properties of Operator $D$ :

$D(u+v) = Du + Dv$
$D(cu) = cDu$ (where $c$ is a constant)
$f(D)$ acts like an algebraic polynomial and can be factored if the coefficients are constants.
$\frac{1}{D}$ represents integration (the inverse operator of $D$ ).

5. Solution of Second Order Homogeneous Linear Differential Equation with Constant Coefficient

A second-order homogeneous LDE with constant coefficients has the form:
$(aD^2 + bD + c)y = 0$
where $a, b,$ and $c$ are real constants and $a \neq 0$ .

The Auxiliary Equation (A.E.)

To solve this, we substitute $y = e^{mx}$ (since derivatives of exponentials are proportional to themselves). This yields the Auxiliary Equation:
$am^2 + bm + c = 0$

Solving this quadratic equation gives two roots, $m_1$ and $m_2$ . The nature of these roots determines the general solution (C.F.):

Case 1: Roots are Real and Distinct ( $m_1 \neq m_2$ )

Solution: $y = c_1 e^{m_1 x} + c_2 e^{m_2 x}$
Where $c_1$ and $c_2$ are arbitrary constants.

Case 2: Roots are Real and Equal ( $m_1 = m_2 = m$ )

Since the roots are the same, $e^{mx}$ is one solution. To find the second linearly independent solution, we multiply by $x$ .
Solution: $y = (c_1 + c_2 x) e^{mx}$

Case 3: Roots are Complex Conjugates ( $m = \alpha \pm i\beta$ )

Using Euler's formula ( $e^{i\theta} = \cos\theta + i\sin\theta$ ), the solution transforms from exponentials into sines and cosines.
Solution: $y = e^{\alpha x} (c_1 \cos \beta x + c_2 \sin \beta x)$

6. Solution of Higher Order Homogeneous Linear Differential Equations with Constant Coefficient

The principles established for second-order equations scale directly to $n$ -th order homogeneous equations:
$(a_n D^n + a_{n-1} D^{n-1} + \dots + a_0)y = 0$

The Auxiliary Equation

The A.E. is an $n$ -th degree polynomial:
$a_n m^n + a_{n-1} m^{n-1} + \dots + a_0 = 0$
This equation has exactly $n$ roots: $m_1, m_2, \dots, m_n$ .

Rules for finding the Complementary Function (C.F.)

Distinct Real Roots ( $m_1, m_2, m_3 \dots$ ):
$y = c_1 e^{m_1 x} + c_2 e^{m_2 x} + c_3 e^{m_3 x} + \dots$
Repeated Real Roots:
- If a root $m$ is repeated 2 times ( $m_1 = m_2 = m$ ):
  $y = (c_1 + c_2 x) e^{mx} + c_3 e^{m_3 x} + \dots$
- If a root $m$ is repeated $k$ times:
  $y = (c_1 + c_2 x + c_3 x^2 + \dots + c_k x^{k-1}) e^{mx}$
Distinct Complex Roots ( $\alpha \pm i\beta$ ):
$y = e^{\alpha x} (c_1 \cos \beta x + c_2 \sin \beta x)$
Repeated Complex Roots:
- If a complex pair $\alpha \pm i\beta$ is repeated twice:
  $y = e^{\alpha x} [(c_1 + c_2 x) \cos \beta x + (c_3 + c_4 x) \sin \beta x]$

Summary of Workflow for Homogeneous LDEs:

Write the equation in operator form: $f(D)y = 0$ .
Write the Auxiliary Equation: $f(m) = 0$ .
Find all roots of the polynomial $f(m) = 0$ .
Apply the rules above to construct the general solution (C.F.).

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Unit 3