Unit 1 — Ordinary Differential Equations of Higher Order

This unit focuses on solving higher-order ordinary differential equations and understanding their applications in engineering problems.

1. Linear Differential Equation of $n$ th Order

A linear differential equation of order $n$ with constant coefficients can be written as:

a_n \frac{d^n y}{dx^n} + a_{n-1} \frac{d^{n-1} y}{dx^{n-1}} + \cdots + a_2 \frac{d^2 y}{dx^2} + a_1 \frac{dy}{dx} + a_0 y = f(x)

where $a_0, a_1, \ldots, a_n$ are constants.

Using the differential operator $D = \dfrac{d}{dx}$ , the equation can also be written as:

F(D)y = f(x)

where

F(D) = a_n D^n + a_{n-1} D^{n-1} + \cdots + a_1 D + a_0

Important Concepts

Complementary Function (CF)
Particular Integral (PI)
Auxiliary Equation
Homogeneous Differential Equation
Non-homogeneous Differential Equation
Linear Differential Equations with Constant Coefficients

Homogeneous Differential Equation

For a homogeneous equation:

F(D)y = 0

the solution is obtained from the auxiliary equation:

F(m) = 0

The roots of the auxiliary equation determine the form of the complementary function.

Case 1 — Distinct Real Roots

If the auxiliary equation has distinct real roots $m_1, m_2, \ldots, m_n$ , then the complementary function is:

y_c = C_1 e^{m_1 x} + C_2 e^{m_2 x} + \cdots + C_n e^{m_n x}

Case 2 — Repeated Real Roots

If $m$ is a repeated root of multiplicity $r$ , then the corresponding solution is:

y_c = e^{mx} \bigl( C_1 + C_2 x + C_3 x^2 + \cdots + C_r x^{r-1} \bigr)

Case 3 — Complex Roots

If the roots are $m = \alpha \pm i\beta$ , then the corresponding solution is:

y_c = e^{\alpha x} \bigl( C_1 \cos \beta x + C_2 \sin \beta x \bigr)

General Solution

For a non-homogeneous differential equation $F(D)y = f(x)$ , the general solution is:

y = y_c + y_p

where:

$y_c$ = Complementary Function
$y_p$ = Particular Integral

2. Simultaneous Linear Differential Equations

Simultaneous differential equations consist of two or more differential equations involving multiple dependent variables.

A general pair can be represented as:

\frac{dx}{dt} = a_1 x + b_1 y + f_1(t)

\frac{dy}{dt} = a_2 x + b_2 y + f_2(t)

These equations are useful for modeling systems where multiple variables interact with each other.

Operator Form

Using $D = \dfrac{d}{dt}$ , a simultaneous system can be expressed as:

F_1(D)x + G_1(D)y = f_1(t)

F_2(D)x + G_2(D)y = f_2(t)

The equations can then be solved by eliminating one of the dependent variables and obtaining a higher-order differential equation for the other.

3. Second-Order Linear Differential Equations with Variable Coefficients

A general second-order linear differential equation with variable coefficients has the form:

a(x) \frac{d^2 y}{dx^2} + b(x) \frac{dy}{dx} + c(x)y = f(x)

where $a(x)$ , $b(x)$ , and $c(x)$ are functions of $x$ .

The corresponding homogeneous equation is:

a(x) \frac{d^2 y}{dx^2} + b(x) \frac{dy}{dx} + c(x)y = 0

The solution depends on the form of the variable coefficients and the available transformation or substitution.

4. Solution by Changing the Independent Variable

A suitable change of independent variable can transform a difficult differential equation into a simpler form.

Suppose $x = \phi(t)$ . Then derivatives with respect to $x$ can be transformed using the chain rule.

For example:

\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{1}{dx/dt} \cdot \frac{dy}{dt}

For the second derivative:

\frac{d^2 y}{dx^2} = \frac{1}{dx/dt} \cdot \frac{d}{dt} \left( \frac{1}{dx/dt} \cdot \frac{dy}{dt} \right)

The appropriate substitution depends on the structure of the differential equation.

5. Method of Variation of Parameters

The variation of parameters method is used to find a particular solution of a non-homogeneous second-order differential equation.

Consider:

\frac{d^2 y}{dx^2} + P(x) \frac{dy}{dx} + Q(x)y = R(x)

First, solve the corresponding homogeneous equation:

\frac{d^2 y}{dx^2} + P(x) \frac{dy}{dx} + Q(x)y = 0

Let its two linearly independent solutions be $y_1(x)$ and $y_2(x)$ .

The complementary function is:

y_c = C_1 y_1 + C_2 y_2

The particular solution is assumed in the form:

y_p = u_1 y_1 + u_2 y_2

where $u_1$ and $u_2$ are functions of $x$ .

The Wronskian is:

W = \begin{vmatrix} y_1 & y_2 \\ y_1' & y_2' \end{vmatrix} = y_1 y_2' - y_2 y_1'

For the normalized equation $y'' + P(x)y' + Q(x)y = R(x)$ , the parameters are:

u_1' = -\frac{y_2 R}{W}, \qquad u_2' = \frac{y_1 R}{W}

Hence:

u_1 = -\int \frac{y_2 R}{W}\, dx, \qquad u_2 = \int \frac{y_1 R}{W}\, dx

The particular solution is therefore:

y_p = u_1 y_1 + u_2 y_2

and the general solution is:

y = y_c + y_p

6. Cauchy-Euler Equation

The Cauchy-Euler equation is an important type of differential equation with variable coefficients.

A general $n$ -th order Cauchy-Euler equation can be written as:

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

For example, a second-order Cauchy-Euler equation is:

a x^2 \frac{d^2 y}{dx^2} + b x \frac{dy}{dx} + c y = f(x)

Homogeneous Cauchy-Euler Equation

For

a x^2 y'' + b x y' + c y = 0

assume a solution of the form $y = x^m$ . Then:

y' = m x^{m-1}, \qquad y'' = m(m-1)x^{m-2}

Substituting these into the differential equation gives:

a\, m(m-1) + b m + c = 0

This is called the auxiliary or indicial equation.

Alternative Transformation

The substitution $x = e^t$ (or equivalently $t = \ln x$ ) can transform a Cauchy-Euler equation into a differential equation with constant coefficients.

Using $t = \ln x$ :

x \frac{dy}{dx} = \frac{dy}{dt}

and

x^2 \frac{d^2 y}{dx^2} = \frac{d^2 y}{dt^2} - \frac{dy}{dt}

This transformation can simplify the solution process.

7. Applications of Differential Equations

Higher-order differential equations are widely used to model engineering systems.

Mechanical Systems

Differential equations can describe the motion of mechanical systems involving displacement, velocity, acceleration, mass, damping, and external forces.

A basic mass-spring system can be represented as:

m \frac{d^2 x}{dt^2} + c \frac{dx}{dt} + k x = F(t)

where:

$m$ = mass
$c$ = damping coefficient
$k$ = spring constant
$F(t)$ = external force
$x(t)$ = displacement

Electrical Circuits

Differential equations can be used to model electrical circuits containing resistors, inductors, and capacitors.

For a series RLC circuit:

L \frac{d^2 q}{dt^2} + R \frac{dq}{dt} + \frac{1}{C} q = E(t)

where:

$L$ = inductance
$R$ = resistance
$C$ = capacitance
$q$ = electric charge
$E(t)$ = applied voltage

Since current is $i = \dfrac{dq}{dt}$ , the equation can also be expressed in terms of current.

Vibrations

Differential equations are used to analyze free and forced vibrations.

A simple undamped vibration system can be represented as:

m \frac{d^2 x}{dt^2} + k x = 0

The natural angular frequency is:

\omega_n = \sqrt{\frac{k}{m}}

Dynamic Systems

Differential equations are used to represent the behavior of dynamic systems over time.

A general linear system can be represented as:

a_n y^{(n)} + a_{n-1} y^{(n-1)} + \cdots + a_1 y' + a_0 y = f(t)

Physical Modeling

Differential equations are also used to model physical phenomena such as:

Motion
Vibrations
Electrical systems
Heat transfer
Fluid flow
Control systems
Dynamic processes

Key Formulas

Linear Differential Equation

F(D)y = f(x)

General Solution

y = y_c + y_p

Auxiliary Equation

F(m) = 0

Cauchy-Euler Equation

a_n x^n y^{(n)} + a_{n-1} x^{n-1} y^{(n-1)} + \cdots + a_1 x y' + a_0 y = f(x)

Variation of Parameters

u_1' = -\frac{y_2 R}{W}, \qquad u_2' = \frac{y_1 R}{W}

Wronskian

W = y_1 y_2' - y_2 y_1'

Cauchy-Euler Substitution

t = \ln x

Mechanical Vibration Equation

m x'' + c x' + k x = F(t)

RLC Circuit Equation

L q'' + R q' + \frac{1}{C} q = E(t)

What You'll Learn

By completing this unit, you will be able to:

Solve linear differential equations of higher order.
Construct auxiliary equations.
Find complementary functions.
Calculate particular integrals.
Solve homogeneous and non-homogeneous equations.
Solve simultaneous linear differential equations.
Work with variable-coefficient differential equations.
Apply changes of independent variables.
Use the method of variation of parameters.
Solve Cauchy-Euler differential equations.
Apply differential equations to mechanical systems.
Model electrical circuits using differential equations.
Analyze vibration and dynamic systems.

Key Takeaway

This unit builds the foundation for solving higher-order differential equations and applying them to real-world engineering systems such as mechanical vibrations, electrical circuits, and dynamic models.

Unit 1 — Ordinary Differential Equations of Higher Order

1. Linear Differential Equation of nnnth Order

Important Concepts

Homogeneous Differential Equation

Case 1 — Distinct Real Roots

Case 2 — Repeated Real Roots

Case 3 — Complex Roots

General Solution

2. Simultaneous Linear Differential Equations

Operator Form

3. Second-Order Linear Differential Equations with Variable Coefficients

4. Solution by Changing the Independent Variable

5. Method of Variation of Parameters

6. Cauchy-Euler Equation

Homogeneous Cauchy-Euler Equation

Alternative Transformation

7. Applications of Differential Equations

Mechanical Systems

Electrical Circuits

Vibrations

Dynamic Systems

Physical Modeling

Key Formulas

Linear Differential Equation

General Solution

Auxiliary Equation

Cauchy-Euler Equation

Variation of Parameters

Wronskian

Cauchy-Euler Substitution

Mechanical Vibration Equation

RLC Circuit Equation

What You'll Learn

Key Takeaway

1. Linear Differential Equation of $n$ th Order