Unit 3 - Notes

MTH174 6 min read

Unit 3: Linear differential equation-II

Introduction to Non-Homogeneous Linear Differential Equations

A general non-homogeneous linear differential equation of order $n$ with constant coefficients is written as:
$\frac{d^n y}{dx^n} + a_1 \frac{d^{n-1} y}{dx^{n-1}} + \dots + a_n y = X(x)$
where $a_1, a_2, \dots, a_n$ are constants and $X(x)$ is a function of $x$ .

The complete solution to this equation consists of two parts:
$y = \text{Complementary Function (C.F.)} + \text{Particular Integral (P.I.)}$
$y = y_c + y_p$

Complementary Function ( $y_c$ ): The general solution of the associated homogeneous equation (where $X(x) = 0$ ).
Particular Integral ( $y_p$ ): Any specific solution to the non-homogeneous equation.

1. Operator Method for Non-Homogeneous LDEs with Constant Coefficients

The differential operator is denoted by $D = \frac{d}{dx}$ , $D^2 = \frac{d^2}{dx^2}$ , etc.
The equation can be written as $f(D)y = X(x)$ , where $f(D) = D^n + a_1 D^{n-1} + \dots + a_n$ .

The Particular Integral (P.I.) is evaluated as:
$\text{P.I.} = \frac{1}{f(D)} X(x)$

Standard Rules for Finding P.I.

Rule 1: When $X(x) = e^{ax}$
$\text{P.I.} = \frac{1}{f(D)} e^{ax} = \frac{1}{f(a)} e^{ax}, \quad \text{provided } f(a) \neq 0$
Failure Case: If $f(a) = 0$ , then $a$ is a root of the auxiliary equation. Multiply the numerator by $x$ and differentiate the denominator with respect to $D$ :
$\text{P.I.} = x \frac{1}{f'(a)} e^{ax}, \quad \text{provided } f'(a) \neq 0$
If $f'(a) = 0$ , repeat the process: multiply by $x^2$ and use $f''(a)$ .

Rule 2: When $X(x) = \sin(ax)$ or $\cos(ax)$
Substitute $D^2 = -a^2$ in $f(D^2)$ .
$\text{P.I.} = \frac{1}{f(D^2)} \sin(ax) = \frac{1}{f(-a^2)} \sin(ax), \quad \text{provided } f(-a^2) \neq 0$
Failure Case: If $f(-a^2) = 0$ , multiply by $x$ and differentiate the denominator w.r.t $D$ :
$\text{P.I.} = x \frac{1}{f'(D)} \sin(ax)$

Rule 3: When $X(x) = x^m$ (Polynomial in $x$ )
$\text{P.I.} = \frac{1}{f(D)} x^m = [f(D)]^{-1} x^m$

Factor out the lowest degree term from $f(D)$ to make it of the form $1 \pm \phi(D)$ .
Expand $[1 \pm \phi(D)]^{-1}$ using the binomial theorem up to the power $m$ (since $D^{m+1}(x^m) = 0$ ).
Apply the operator terms to $x^m$ .

Rule 4: When $X(x) = e^{ax} \cdot V(x)$ (where $V(x)$ is any function of $x$ )
Use the Shift Theorem:
$\text{P.I.} = \frac{1}{f(D)} [e^{ax} V(x)] = e^{ax} \frac{1}{f(D+a)} V(x)$
After shifting $e^{ax}$ to the left, operate $\frac{1}{f(D+a)}$ on $V(x)$ using the appropriate rule.

Rule 5: When $X(x) = x \cdot V(x)$
$\text{P.I.} = \frac{1}{f(D)} [x V(x)] = \left[ x - \frac{f'(D)}{f(D)} \right] \frac{1}{f(D)} V(x)$

2. Method of Undetermined Coefficients

This method is used to find the P.I. when the non-homogeneous term $X(x)$ (or $R(x)$ ) is a polynomial, exponential, sine, cosine, or a finite sum/product of these. It involves guessing the form of the P.I. with unknown coefficients and determining them by substituting back into the differential equation.

Step-by-Step Procedure

Find the Complementary Function ( $y_c$ ) by solving the homogeneous equation.
Choose a Trial Solution ( $y_p$ ) based on $X(x)$ from the table below.
Modification Rule: If any term in the assumed trial solution is already present in the Complementary Function, multiply the entire trial solution by $x$ (or $x^2$ , $x^3$ , etc., until no term duplicates a term in $y_c$ ).
Differentiate the trial solution as required by the order of the ODE.
Substitute $y_p, y_p', y_p'' \dots$ into the original differential equation.
Equate coefficients of like terms on both sides to solve for the undetermined constants.

Table of Trial Solutions

Non-homogeneous Term $X(x)$	Assumed Trial Solution $y_p$
$k$ (constant)	$A$
$x^n$ (polynomial of degree $n$ )	$A_n x^n + A_{n-1} x^{n-1} + \dots + A_1 x + A_0$
$e^{ax}$	$A e^{ax}$
$\sin(ax)$ or $\cos(ax)$	$A \sin(ax) + B \cos(ax)$
$e^{ax} \sin(bx)$ or $e^{ax} \cos(bx)$	$e^{ax} (A \sin(bx) + B \cos(bx))$

3. Method of Variation of Parameters

This is a powerful, general method for finding the Particular Integral of linear differential equations. Unlike the Method of Undetermined Coefficients, it applies to any function $X(x)$ , including $\tan x$ , $\sec x$ , $\ln x$ , etc.

For a Second-Order Linear Differential Equation

Consider the standard form:
$\frac{d^2 y}{dx^2} + P \frac{dy}{dx} + Q y = R(x)$
where $P$ and $Q$ can be constants or functions of $x$ .

Procedure:

Find the Complementary Function ( $y_c$ ). Let the roots give two linearly independent solutions $y_1(x)$ and $y_2(x)$ .
$y_c = c_1 y_1 + c_2 y_2$
Calculate the Wronskian ( $W$ ) of $y_1$ and $y_2$ :
$W = \begin{vmatrix} y_1 & y_2 \\ y_1' & y_2' \end{vmatrix} = y_1 y_2' - y_2 y_1'$
(Note: $W$ must be non-zero for linearly independent solutions).
Assume the Particular Integral is of the form:
$y_p = u(x) y_1(x) + v(x) y_2(x)$
The unknown functions $u(x)$ and $v(x)$ are given by the formulas:
$u(x) = -\int \frac{y_2 \cdot R(x)}{W} dx$
$v(x) = \int \frac{y_1 \cdot R(x)}{W} dx$
The complete solution is $y = y_c + y_p$ .

4. Solution of Euler-Cauchy Equation

The Euler-Cauchy equation (or Cauchy linear differential equation) is a linear differential equation with variable coefficients of a specific form. The degree of the independent variable $x$ matches the order of the derivative.

Standard Form

$x^n \frac{d^n y}{dx^n} + a_1 x^{n-1} \frac{d^{n-1} y}{dx^{n-1}} + \dots + a_{n-1} x \frac{dy}{dx} + a_n y = f(x)$
where $a_1, a_2, \dots, a_n$ are constants.

Method of Solution

To solve this, we convert the variable coefficient differential equation into a constant coefficient differential equation by changing the independent variable.

Substitution:
Let $x = e^z$ , which implies $z = \ln(x)$ .
Let the differential operator with respect to $z$ be $D \equiv \frac{d}{dz}$ .
Transformation of Derivatives:
Using the chain rule, we can show:
- $x \frac{dy}{dx} = Dy$
- $x^2 \frac{d^2 y}{dx^2} = D(D-1)y$
- $x^3 \frac{d^3 y}{dx^3} = D(D-1)(D-2)y$
- ...and so on, where $D = \frac{d}{dz}$ .
Substituting into the Equation:
Replace the terms $x^k \frac{d^k y}{dx^k}$ in the original ODE with the $D$ operator forms.
Also, replace the right-hand side $f(x)$ with $f(e^z)$ .
Solve the Resulting Equation:
The equation is now a linear differential equation with constant coefficients in terms of variables $y$ and $z$ .
Solve it using the standard Operator Method or Method of Undetermined Coefficients to find $y$ as a function of $z$ :
$y(z) = \text{C.F.}(z) + \text{P.I.}(z)$
Back-Substitution:
Finally, replace $z$ with $\ln(x)$ and $e^z$ with $x$ to obtain the complete solution in terms of the original variables $y$ and $x$ .

Unit 2

Unit 4