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?

<p>An equation involving functions and their derivatives (A)</p> Signup and view all the answers

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

<p>A solution must contain at least one arbitrary constant (B)</p> Signup and view all the answers

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

<p>The highest coefficient in the equation (B)</p> Signup and view all the answers

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

<p>To model diverse physical, technical, or biological processes (C)</p> Signup and view all the answers

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

<p>Homogeneous constant coefficient method (C)</p> Signup and view all the answers

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

<p>They are the same. (B)</p> Signup and view all the answers

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

<p>The general solution involves only exponential functions. (D)</p> Signup and view all the answers

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

<p>The form is $e^{m_1 x}, x e^{m_1 x}, x^2 e^{m_1 x}, ...$. (D)</p> Signup and view all the answers

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

<p>There are one double root and one distinct root. (B)</p> Signup and view all the answers

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

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

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

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

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

<p>Three solutions are provided. (C)</p> Signup and view all the answers

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

<p>An oscillatory solution involving sine and cosine. (B)</p> Signup and view all the answers

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

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

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

<p>It is a multiple root of the same order. (B)</p> Signup and view all the answers

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

<p>$m^2 + 4m + 13 = 0$ (C)</p> 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?

<p>$1/3$ (D)</p> Signup and view all the answers

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

<p>Constants derived from the initial conditions (C)</p> Signup and view all the answers

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

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

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

<p>It introduces oscillation in the solution. (A)</p> Signup and view all the answers

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

<p>$3$ (A)</p> Signup and view all the answers

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

<p>$a_0 x y'' + a_1 x y' + a_2 x y = 0$ (D)</p> Signup and view all the answers

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

<p>$D = d/dx$ (B)</p> Signup and view all the answers

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

<p>$r(x)$ is a function of x. (D)</p> Signup and view all the answers

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

<p>The third derivative of f. (C)</p> Signup and view all the answers

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

<p>$L = a_0 x D^n + a_1 x D^{n-1} + ... + a_n x$ (B), $L = a_0 x D^n + a_1 D^{n-1} + a_n D^0$ (D)</p> Signup and view all the answers

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

<p>$y = e^{mx}$ (D)</p> Signup and view all the answers

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

<p>y is a solution to the differential equation. (B)</p> Signup and view all the answers

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

<p>They involve both derivatives and functions of x. (A)</p> Signup and view all the answers

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

<p>They are constants. (A)</p> Signup and view all the answers

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

<p>Application of the polynomial operator P on the function y. (C)</p> Signup and view all the answers

Flashcards

Differential Equation

An equation relating functions and their derivatives.

Solution (Primitive)

Equation without derivatives, satisfying a differential equation.

General Solution

A solution with arbitrary constants.

Particular Solution

A solution derived from the general solution with specific values for constants.

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Homogeneous Equation

A differential equation with a right-hand side of zero.

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Linear Ordinary Differential Equation

A differential equation where the dependent variable and its derivatives appear to the first power.

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Higher Order ODE

Differential equation with derivatives of order 2 or higher.

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Constant Coefficients

Coefficients next to the derivatives (e.g., a0, a1) in a differential equation that are constants.

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Characteristic Equation

An algebraic equation derived from a differential equation, enabling finding solutions.

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Real and Distinct Roots

Roots of the characteristic equation which are separate real numbers

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Multiple Roots

Roots of the characteristic equation that repeat.

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Complex Roots

Roots of the characteristic equation with real and imaginary parts (p±iq).

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General Solution (Distinct Roots)

Sum of exponential functions where each exponential corresponds to a root.

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Multiple Root Solution

The solution for repeated roots includes additional terms involving x times the basic exponential function

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Order of a Differential Equation

Highest derivative present in the given differential equation.

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Solution of a second-order differential equation with complex roots

Written as 𝑦 𝑥 = 𝑒 𝑝𝑥 [𝑐1 cos 𝑞𝑥 + 𝑐2 𝑠𝑖𝑛 𝑞𝑥], where p and q are the real and imaginary parts of the complex root, and c1 and c2 are constants determined from initial conditions.

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Characteristic equation for a second-order differential equation

A quadratic equation, typically in the form 𝑚2 + 4𝑚 + 13 = 0, derived from the differential equation. It's used to find the roots (solutions).

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General solution for second-order differential equation

The most general form of the solution including both possible real and complex component solutions.

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Multiple complex roots (differential equation)

If the characteristic equation has a repeating complex root, the general solution includes additional linearly independent solutions involving the variable x.

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Initial value problem

A differential equation combined with initial conditions (values for y and y' at time=0)

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Complementary Function (C.F.)

Another name for the solution of a homogeneous part, in a differential equation

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Complex roots in a differential equation

Roots of the characteristic equation, which are of the form (p ± iq), where p and q are real numbers.

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Second-order homogeneous equation

A differential equation in the form 𝑎0𝑥𝑦′′ + 𝑎1𝑥𝑦′ + 𝑎2𝑥𝑦 = 0, where 𝑎0𝑥 ≠ 0.

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Second-order non-homogeneous equation

A differential equation in the form 𝑎0𝑥𝑦′′ + 𝑎1𝑥𝑦′ + 𝑎2𝑥𝑦 = 𝑟(𝑥), where 𝑎0𝑥 ≠ 0.

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Differential Operator D

A notation that represents the derivative operation with respect to x. D = d/dx

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D^n f(x)

Represents the nth derivative of the function f(x).

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Linear differential equation with constant coefficients

A differential equation where the coefficients of the derivatives are constants and it is linear (no products or powers of the dependent variable).

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Operator L

A differential operator defined as a polynomial in the operator D L = a0𝑥𝐷𝑛 + a1𝑥𝐷n−1 + ⋯ + an−1𝑥𝐷 + an.

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nth order homogeneous linear equation with constant coefficients

A differential equation in the form 𝑎0𝑦^(𝑛) + 𝑎1𝑦^(𝑛−1) + ... + 𝑎𝑛−1𝑦′ + 𝑎𝑛𝑦 = 0,with constant coefficients.

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Solution y = e^(mx)

A possible solution form for a homogeneous linear differential equation with constant coefficients.

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Solution Form y = e^(mx)

A proposed solution where y represents the function in question and m is a constant.

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Polynomial in D

An expression that's a combination of powers of the differential operator D.

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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(dn-1y/dxn-1) + ... + an-1(dy/dx) + any = 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=emx leads to the characteristic equation: a₀mn + a₁mn-1 + ... + an-1m + an = 0

Examples of Solutions

  • Real and distinct roots:
    • If the characteristic equation has n distinct real roots (m₁, m₂, ..., mn), the general solution is y(x) = C₁em₁x + C₂em₂x + ... + Cnemnx.
  • 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)]epx.

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.

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