Mathematics 1: Homogeneous ODEs

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Questions and Answers

What is a linear ordinary differential equation of order n characterized by?

Having arbitrary constants in its solution

Only dependent variables present in the equation

The absence of derivatives in its form

A constant coefficient multiplied by the highest derivative (correct)

How is a particular solution of a differential equation defined?

It is obtained from the general solution by assigning specific values to arbitrary constants (correct)

It is derived by ignoring the independent variable

It includes several arbitrary constants

It has constant coefficients only

What distinguishes a homogeneous linear ordinary differential equation from a non-homogeneous one?

It functions only when all coefficients are variables

The independent variable is not included

The presence of any function on the right side of the equation (correct)

The coefficients of derivatives must be null

Which of the following best describes the concept of a differential equation?

An equation involving functions and their derivatives Signup and view all the answers

Which statement regarding the solution of a differential equation is true?

A solution must contain at least one arbitrary constant Signup and view all the answers

What is indicated by the symbol $a_n(x)y$ in a linear ordinary differential equation?

The highest coefficient in the equation Signup and view all the answers

What is the primary use of differential equations across various disciplines?

To model diverse physical, technical, or biological processes Signup and view all the answers

Which of the following is NOT a method commonly used for solving first-order differential equations?

Homogeneous constant coefficient method Signup and view all the answers

What is the relationship between the degree of the characteristic equation and the order of the differential equation?

They are the same. Signup and view all the answers

What does it indicate if the characteristic equation has real and distinct roots?

The general solution involves only exponential functions. Signup and view all the answers

In the case of a multiple root in the characteristic equation, how is the general solution formulated?

The form is $e^{m_1 x}, x e^{m_1 x}, x^2 e^{m_1 x}, ...$. Signup and view all the answers

Given the characteristic equation $m^3 - 3m - 2 = 0$, which of the following sets of roots is correct?

There are one double root and one distinct root. Signup and view all the answers

Which term represents the general solution if the characteristic equation has complex conjugate roots?

$c_1 e^{(p+iq)x} + c_2 e^{(p-iq)x}$. Signup and view all the answers

What is the general solution for the given differential equation with distinct real roots $m=1, -2, 3$?

$y(x) = Ae^{x} + Be^{-2x} + Ce^{3x}$. Signup and view all the answers

How many linearly independent solutions are provided by a triple root in the characteristic equation?

Three solutions are provided. Signup and view all the answers

What type of solution would you expect if all the roots of the characteristic equation are imaginary?

An oscillatory solution involving sine and cosine. Signup and view all the answers

What is the general form of the solution for the differential equation with complex roots?

$y(x) = e^{px}[C_1 ext{cos}(qx) + C_2 ext{sin}(qx)]$ Signup and view all the answers

In the case of multiple complex roots $p + iq$, what can be stated about $p - iq$?

It is a multiple root of the same order. Signup and view all the answers

What is the characteristic equation for the differential equation $y'' + 4y' + 13y = 0$?

$m^2 + 4m + 13 = 0$ Signup and view all the answers

For the initial value problem $y'' + 4y' + 13y = 0$, what is the value of $B$ after applying initial conditions?

$1/3$ Signup and view all the answers

What do $c_1$ and $c_2$ represent in the final form of the solution $y(x) = e^{px}[c_1 ext{cos}(qx) + c_2 ext{sin}(qx)]$?

Constants derived from the initial conditions Signup and view all the answers

What is the form of the solution for the differential equation $y^{iv} + 32y'' + 256y = 0$?

$y(x) = Ax + B ext{cos}(4x) + C x + D ext{sin}(4x)$ Signup and view all the answers

What effect does an imaginary part $q$ have in the solutions involving complex roots?

It introduces oscillation in the solution. Signup and view all the answers

What is the value of $q$ in the context of the characteristic roots $m = -2 ext{±} 3i$?

$3$ Signup and view all the answers

What is the general form of a second order homogeneous equation?

$a_0 x y'' + a_1 x y' + a_2 x y = 0$ Signup and view all the answers

Which operator is used to denote differentiation with respect to x?

$D = d/dx$ Signup and view all the answers

In a non-homogeneous second order equation, what does the right side of the equation represent?

$r(x)$ is a function of x. Signup and view all the answers

What does the notation $D^3 f$ indicate?

The third derivative of f. Signup and view all the answers

Which of the following is the proper representation of a linear differential operator of order n?

$L = a_0 x D^n + a_1 x D^{n-1} + ... + a_n x$ Signup and view all the answers

What is the solution form proposed for the nth order homogeneous linear equation with constant coefficients?

$y = e^{mx}$ Signup and view all the answers

In the context of the operator L, what does $L y = 0$ signify?

y is a solution to the differential equation. Signup and view all the answers

Which of the following correctly describes the terms in the operator L for the equation $L y = 0$?

They involve both derivatives and functions of x. Signup and view all the answers

What is a characteristic of the coefficients $a_i$ in the linear second order constant coefficient equations?

They are constants. Signup and view all the answers

What does the expression $P(D) y$ in the context of the operator L represent?

Application of the polynomial operator P on the function y. Signup and view all the answers

Study Notes

Lecture 25: Mathematics 1 (15B11MA111)

Course code: 15B11MA111
Module: Ordinary Differential Equations
Topic: Homogeneous constant coefficient ODEs
Reference: R.K. Jain and S.R.K. Iyenger, "Advanced Engineering Mathematics" fifth edition, Narosa publishing house, 2016

Differential Equations

A differential equation relates one or more functions and their derivatives.
The study of differential equations is a broad field in mathematics, physics, and engineering.
Differential equations are crucial for modeling physical, technical, and biological processes, from celestial motion to interactions between neurons.

Types Of Differential Equations

Ordinary Differential Equations (ODE): An ODE contains one or more functions of one independent variable and their derivatives.
- Example ODEs:
  - (dy)/(dx) = 1 + x^2
  - d²y/dx² - 2 dy/dx - 8y = 0
  - (1+ (dy/dx)^2)^3/2=k (d^2 y/dx^2)
Partial Differential Equations (PDE): A PDE involves unknown multivariable functions and their partial derivatives. It contrasts with ODEs, which deal with single-variable functions.
- Example PDEs:
  - x (∂u/∂x) + y (∂u/∂y) = nu
  - (∂²z/∂x∂y) = (∂z/∂y)

Order and Degree of a Differential Equation

Order: The order of a differential equation is the order of the highest derivative present.
- Example orders:
  - 2 (for examples 1 and 2 above)
  - 2 (for example 3 above)
Degree: The degree is the power of the highest order derivative after removing radicals and fractions.
- Example degrees:
  - 1 (for equations 1 and 2)
  - 2 (for equation 3)

Linear Differential Equation

A linear differential equation has these properties:
- The dependent function and its derivatives are in the first degree.
- No product of the function and its derivatives exist.
- No transcendental functions (trig, log, etc.) of the function or its derivatives appear.

Solution of a Differential Equation

A solution satisfies the differential equation and is free of derivatives.
The general solution contains arbitrary constants.
A particular solution is obtained from the general solution by assigning specific values to the arbitrary constants.

Higher Order Linear Differential Equations with Constant Coefficients

A linear ODE of order n with constant coefficients has the form: a₀(dⁿy/dxⁿ) + a₁(d^n-1y/dx^n-1) + ... + a_n-1(dy/dx) + a_ny = r(x)
If r(x) = 0, the equation is homogeneous; otherwise it's non-homogeneous.

Solution of Higher Order Homogenous Linear Equations with Constant Coefficients

Attempting a solution in the form y=e^mx leads to the characteristic equation: a₀mⁿ + a₁m^n-1 + ... + a_n-1m + a_n = 0

Examples of Solutions

Real and distinct roots:
- If the characteristic equation has n distinct real roots (m₁, m₂, ..., m_n), the general solution is y(x) = C₁e^m₁x + C₂e^m₂x + ... + C_ne^m_nx.
Multiple real roots:
- Repeated/multiple roots are handled by adding linearly independent solutions like xy₁, x²y₁, etc. For a root with multiplicity 'r', r-1 additional linearly independent terms will appear
Complex roots:
- Complex roots (conjugate pairs) appear as [C₁ cos(qx) + C₂ sin(qx)]e^px.

Additional Notes

Initial value problems require finding specific solutions that meet given initial conditions.
Complementary Function (C.F.): The solution of the homogeneous part of a differential equation.

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Description

Test your knowledge on homogeneous constant coefficient ordinary differential equations as part of Mathematics 1 (Course 15B11MA111). This quiz will cover essential concepts detailed in the 'Advanced Engineering Mathematics' textbook. Prepare to solve various ODE problems and enhance your understanding of differential equations.