Unit 3 - Notes

MTH166

Unit 3: Linear differential equation

1. Fundamentals of Linear Differential Equations

A linear differential equation of order $n$ with constant coefficients is of the form:

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

Where

a_0, a_1, \dots, a_n

are constants and

X(x)

is a function of

x

The General Solution consists of two parts:

y = y_c + y_p

Complementary Function ( $y_c$ or C.F.): The solution to the homogeneous part (LHS = 0).
Particular Integral ( $y_p$ or P.I.): A specific solution satisfying the non-homogeneous equation (LHS = $X(x)$ ).

Finding the Complementary Function (C.F.)

Let the auxiliary equation (A.E.) be $f(m) = 0$ by replacing $\frac{d}{dx}$ with $m$ .

Nature of Roots ( $m_1, m_2, \dots$ )	Form of C.F.
Real and Distinct ( $m_1 \neq m_2$ )	$y_c = c_1 e^{m_1 x} + c_2 e^{m_2 x}$
Real and Equal ( $m_1 = m_2 = m$ )	$y_c = (c_1 + c_2 x) e^{mx}$
Complex Conjugates ( $\alpha \pm i\beta$ )	$y_c = e^{\alpha x} (c_1 \cos \beta x + c_2 \sin \beta x)$
Repeated Complex Roots ( $\alpha \pm i\beta, \alpha \pm i\beta$ )	$y_c = e^{\alpha x} [(c_1 + c_2 x) \cos \beta x + (c_3 + c_4 x) \sin \beta x]$

2. Solution by Operator Method (Finding P.I.)

Denote the differential operator $D = \frac{d}{dx}$ . The equation is written as $f(D)y = X$ .
The Particular Integral is given by:

y_p = \frac{1}{f(D)} X

Case I: $X = e^{ax}$

Rule: Replace $D$ with $a$ .
$y_p = \frac{1}{f(a)} e^{ax}, \quad \text{provided } f(a) \neq 0$
Failure Case ( $f(a) = 0$ ):
If $f(a) = 0$ , differentiate the denominator with respect to $D$ and multiply numerator by $x$ .
$y_p = x \cdot \frac{1}{f'(a)} e^{ax}$
If $f'(a)$ is still 0, repeat the process ( $x^2 \cdot \frac{1}{f''(a)} e^{ax}$ ).

Case II: $X = \sin(ax)$ or $\cos(ax)$

Rule: Replace $D^2$ with $-a^2$ . (Do not replace $D$ , only $D^2$ ).
$y_p = \frac{1}{f(D^2)} \sin(ax) = \frac{1}{f(-a^2)} \sin(ax), \quad \text{provided } f(-a^2) \neq 0$
Note: If the denominator contains terms like $D + b$ after substitution, multiply numerator and denominator by the conjugate ( $D - b$ ) to create a $D^2$ term in the denominator.
Failure Case ( $f(-a^2) = 0$ ):
$y_p = x \cdot \frac{1}{f'(D)} \sin(ax)$

Case III: $X = x^m$ (Polynomial)

Rule: Expand using Binomial Theorem.
1. Factor out the lowest degree term from $f(D)$ to make the form $[1 \pm \phi(D)]$ .
2. Bring it to the numerator as $[1 \pm \phi(D)]^{-1}$ .
3. Expand using:
  - $(1-D)^{-1} = 1 + D + D^2 + D^3 + \dots$
  - $(1+D)^{-1} = 1 - D + D^2 - D^3 + \dots$
4. Operate on $x^m$ until derivatives become zero ( $D^{m+1}x^m = 0$ ).

Case IV: $X = e^{ax} V(x)$ (Exponential Shift)

Rule: Shift $e^{ax}$ outside the operator and replace $D$ with $(D+a)$ .
$y_p = \frac{1}{f(D)} e^{ax} V(x) = e^{ax} \frac{1}{f(D+a)} V(x)$
Then evaluate $\frac{1}{f(D+a)} V(x)$ using Case II or Case III methods depending on $V(x)$ .

Case V: $X = x V(x)$

Rule:
$y_p = \frac{1}{f(D)} x V(x) = \left[ x - \frac{f'(D)}{f(D)} \right] \frac{1}{f(D)} V(x)$

3. Method of Undetermined Coefficients

This method involves "guessing" the form of the particular solution based on the form of $X(x)$ and determining the unknown coefficients by substituting into the original DE.

Constraint: Applicable only when $X(x)$ is a polynomial, exponential, sine/cosine, or sums/products of these.

Selection Table for Trial Solution ( $y_p$ )

Form of $X(x)$	Trial Solution ( $y_p$ )
$k e^{ax}$	$A e^{ax}$
$k \sin(ax)$ or $k \cos(ax)$	$A \cos(ax) + B \sin(ax)$
$k x^2$ (Polynomial of degree $n$ )	$Ax^2 + Bx + C$ (Polynomial of same degree)
$k e^{ax} \cos(bx)$	$e^{ax} (A \cos bx + B \sin bx)$

The Modification Rule

If any term in the chosen Trial Solution matches a term in the Complementary Function ( $y_c$ ):

Multiply the trial solution by $x$ .
If it still matches, multiply by $x^2$ .

Steps:

Find $y_c$ .
Select appropriate $y_p$ with unknown coefficients ( $A, B, \dots$ ).
Apply modification rule if necessary.
Differentiate $y_p$ to find $y_p', y_p''$ .
Substitute these into the LHS of the DE.
Equate coefficients of like terms on LHS and RHS to solve for $A, B, \dots$ .

4. Method of Variation of Parameters

This is a general method usually used for second-order equations when the Operator Method is difficult or $X(x)$ is not a standard form (e.g., $\tan x, \sec x$ ).

Standard Form: $y'' + P y' + Q y = R(x)$

Steps:

Find C.F.: Let $y_c = c_1 y_1(x) + c_2 y_2(x)$ .
Here, $y_1$ and $y_2$ are independent solutions of the homogeneous equation.
Calculate Wronskian ( $W$ ):
$W = \begin{vmatrix} y_1 & y_2 \\ y_1' & y_2' \end{vmatrix} = y_1 y_2' - y_2 y_1'$
Note: $W$ must not be zero.
Assume P.I.:
$y_p = u(x) y_1 + v(x) y_2$
Calculate $u(x)$ and $v(x)$ :
$u(x) = - \int \frac{y_2 \cdot R(x)}{W} dx$
$v(x) = \int \frac{y_1 \cdot R(x)}{W} dx$
Final Solution: $y = c_1 y_1 + c_2 y_2 + u y_1 + v y_2$

5. Euler-Cauchy Equation

Also known as the linear equidimensional equation. It has variable coefficients where the power of $x$ matches the order of the derivative.

General 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 y = X(x)

Method of Solution: Transform into a linear DE with constant coefficients.

Substitution:
Let $x = e^z$ $\implies$ $z = \ln x$ .
Operator Transformation:
Let and .
- $x \frac{dy}{dx} = x D y = \theta y$
- $x^2 \frac{d^2y}{dx^2} = x^2 D^2 y = \theta (\theta - 1) y$
- $x^3 \frac{d^3y}{dx^3} = x^3 D^3 y = \theta (\theta - 1) (\theta - 2) y$
Substitute and Solve:
- Replace terms in the original equation with $\theta$ terms.
- Replace $X(x)$ on RHS with function of $z$ (replace $x$ with $e^z$ ).
- Solve the new constant coefficient DE for $y(z)$ .
Back-Substitution:
Replace $e^z$ with $x$ and $z$ with $\ln x$ in the final solution to get $y(x)$ .

6. Simultaneous Differential Equations

Solving a system of linear differential equations involving two dependent variables (say $x(t), y(t)$ ) and one independent variable ( $t$ ).

Form:

$f_1(D)x + g_1(D)y = T_1(t)$
$f_2(D)x + g_2(D)y = T_2(t)$
Where $D = \frac{d}{dt}$ .

Method (Operator / Elimination Method):
This is analogous to solving simultaneous algebraic equations.

Write in Notation: Express the equations using operator $D$ .
Eliminate one variable:
- To eliminate $y$ , multiply eq(1) by $g_2(D)$ and eq(2) by $g_1(D)$ .
- Subtract the equations. This yields a higher-order linear DE entirely in terms of $x$ and $t$ .
Solve for first variable: Solve the resulting DE for $x(t)$ using C.F. and P.I. methods.
Find the second variable:
- Option A: Repeat the elimination process for $x$ to find $y$ .
- Option B (Preferred): Substitute the known solution of $x(t)$ into one of the original equations (usually the simpler one) to solve for $y(t)$ .
Constants: Ensure the number of arbitrary constants matches the total order of the system. If Option A is used, you must substitute back to ensure constants in $x$ and $y$ are related correctly. Option B handles this automatically.

Unit 3 - Notes

Table of Contents

Unit 3: Linear differential equation

1. Fundamentals of Linear Differential Equations

Finding the Complementary Function (C.F.)

2. Solution by Operator Method (Finding P.I.)

Case I:

Case II: or

Case III: (Polynomial)

Case IV: (Exponential Shift)

Case V: