Differential Equations - Homogenous Domain- Definition & an Example

  • Differential Equation:

    • An equation involving derivatives of an unknown function is called a differential equation.
    • It represents the relationship between a function and its derivatives.

  • Homogenous Differential Equation:

    • A homogeneous differential equation is a differential equation in which all the terms containing the dependent variable and its derivatives are of the same degree.
    • It can be expressed as:
      • $ F(x, y, y’, y’’, …, y^n) = 0 $

  • Order of a Differential Equation:

    • The power of the highest derivative in a differential equation is called its order.
    • Example:
      • $ y’’ - 5x^2 y’ + 2xy = 0 $ is a second-order differential equation.

  • Degree of a Differential Equation:

    • The power of the highest power of the highest derivative in a differential equation is called its degree.
    • Example:
      • $ x^2y’’ + 3xy’ + y = 0 $ is a second-degree differential equation.

  • Homogeneous First-Order Differential Equation:

    • A homogeneous first-order differential equation can be written as:
      • $ \frac{dy}{dx} = f\left(\frac{y}{x}\right) $
    • Example:
      • $ x\frac{dy}{dx} - y = 2x $ is a homogeneous first-order differential equation.

  • Method to solve Homogeneous First-order Differential Equation:

    1. Substitute $ y = vx $ and differentiate.
    2. Express $ \frac{dy}{dx} $ in terms of $ v $ and $ x $ .
    3. Replace $ \frac{dy}{dx} $ and $ y $ in the given differential equation.
    4. Solve the resulting equation for $ v $ .
    5. Replace $ v $ with $ \frac{y}{x} $ and solve for $ y $ .

  • Homogeneous Second-Order Differential Equation:

    • A homogeneous second-order differential equation can be written as:
      • $ a \frac{d^2y}{dx^2} + b \frac{dy}{dx} + cy = 0 $
    • Example:
      • $ x^2\frac{d^2y}{dx^2} + 3x\frac{dy}{dx} - y = 0 $ is a homogeneous second-order differential equation.

  • Method to solve Homogeneous Second-order Differential Equation:

    • To solve homogeneous second-order differential equations, we use the auxiliary equation.
    • Steps:
      1. Assume $ y = e^{mx} $ where $ m $ is a constant.
      2. Differentiate $ y $ with respect to $ x $ and substitute in the differential equation.
      3. Solve the resulting auxiliary equation to find the values of $ m $ .
      4. The general solution is of the form $ y = C_1 e^{mx_1} + C_2 e^{mx_2} $ where $ C_1 $ and $ C_2 $ are constants and $ m_1 $ and $ m_2 $ are the solutions of the auxiliary equation.

  • Homogeneous Higher-Order Differential Equation:

    • A homogeneous higher-order differential equation can be written as:
      • $ a_n \frac{d^ny}{dx^n} + a_{n-1} \frac{d^{n-1}y}{dx^{n-1}} + … + a_1 \frac{dy}{dx} + a_0y = 0 $
    • Example:
      • $ x^3\frac{d^3y}{dx^3} - 2x^2\frac{d^2y}{dx^2} + 4x\frac{dy}{dx} - 6y = 0 $ is a homogeneous third-order differential equation.

  • Method to solve Homogeneous Higher-order Differential Equation:

    • The method to solve homogeneous higher-order differential equations is similar to that of second-order differential equations.
    • Use the auxiliary equation method to find the general solution.

  • Homogeneous Differential Equation with Variable Coefficients:

    • A homogeneous differential equation with variable coefficients can be written as:
      • $ a(x) \frac{d^2y}{dx^2} + b(x) \frac{dy}{dx} + c(x)y = 0 $
    • Example:
      • $ x^2\frac{d^2y}{dx^2} - x(1+\ln{x})\frac{dy}{dx} + (1+\ln{x})y = 0 $ is a homogeneous differential equation with variable coefficients.

  • Method to solve Homogeneous Differential Equation with Variable Coefficients:

    • We can solve this type of differential equation using power series method or by finding an integrating factor.
  1. Homogeneous Differential Equation with Variable Coefficients - Power Series Method
  • In the power series method, the solution is assumed to be an infinite power series of the form:
    • $ y = \sum_{n=0}^{\infty} a_n x^n $
  • Substitute the power series solution into the given differential equation.
  • Equate the coefficients of like powers of $ x $ on both sides of the equation.
  • Solve the resulting recurrence relation to find the values of $ a_n $ .
  • Substitute the values of $ a_n $ back into the power series solution to obtain the general solution.
  1. Homogeneous Differential Equation with Variable Coefficients - Finding an Integrating Factor
  • If the differential equation can be written in the form:
    • $ \frac{d}{dx}(p(x) \frac{dy}{dx}) + q(x) y = 0 $
  • The integrating factor is given by:
    • $ I(x) = e^{\int p(x) dx} $
  • Multiply both sides of the differential equation by the integrating factor.
  • The resulting equation can be rewritten in the form:
    • $ \frac{d}{dx}(I(x) \frac{dy}{dx}) + I(x) q(x) y = 0 $
  • This equation can be integrated directly to obtain the general solution.
  1. Homogeneous Differential Equations - Summary and Key Points
  • Homogeneous differential equations only contain terms of the same degree involving the dependent variable and its derivatives.
  • The order of a differential equation is determined by the highest derivative present.
  • The degree of a differential equation is determined by the exponent of the highest derivative.
  • Homogeneous first-order differential equations can be solved using the substitution method.
  • Homogeneous second and higher-order differential equations can be solved using the auxiliary equation method.
  • Homogeneous differential equations with variable coefficients can be solved using the power series method or by finding an integrating factor.
  1. Example 1: Homogeneous First-Order Differential Equation
  • Given $ \frac{dy}{dx} = \frac{2y-x}{x+y} $
  • Substitute $ y = vx $ and differentiate to get $ \frac{dy}{dx} = v + x\frac{dv}{dx} $
  • Substitute in the given equation: $ v + x\frac{dv}{dx} = \frac{2(vx)-x}{x+(vx)} $
  • Simplify and solve for $ v $ : $ v = -\frac{x}{x+2} $
  • Substitute $ v = \frac{y}{x} $ and solve for $ y $ to obtain the general solution.
  1. Example 2: Homogeneous Second-Order Differential Equation
  • Given $ x^2\frac{d^2y}{dx^2} + 3x\frac{dy}{dx} - y = 0 $
  • Assume $ y = e^{mx} $ and differentiate to get $ \frac{d^2y}{dx^2} = m^2 e^{mx} $
  • Substitute in the given equation and simplify to get a quadratic equation in $ m $ .
  • Solve the auxiliary equation to find the values of $ m $ .
  • The general solution is of the form $ y = C_1 e^{m_1 x} + C_2 e^{m_2 x} $ , where $ C_1 $ and $ C_2 $ are constants and $ m_1 $ and $ m_2 $ are the solutions of the auxiliary equation.
  1. Example 3: Homogeneous Differential Equation with Variable Coefficients
  • Given $ x^2\frac{d^2y}{dx^2} - x(1+\ln{x})\frac{dy}{dx} + (1+\ln{x})y = 0 $
  • Rewrite the equation in the form: $ \frac{d}{dx}(p(x) \frac{dy}{dx}) + q(x) y = 0 $
  • Identify the values of $ p(x) $ and $ q(x) $ and calculate the integrating factor $ I(x) $ .
  • Multiply both sides of the equation by the integrating factor and simplify.
  • Integrate the resulting equation to obtain the general solution.
  1. Summary of Methods to Solve Homogeneous Differential Equations
  • Homogeneous first-order differential equations can be solved using the substitution method.
  • Homogeneous second and higher-order differential equations can be solved using the auxiliary equation method.
  • Homogeneous differential equations with variable coefficients can be solved using the power series method or by finding an integrating factor.
  • It is important to identify the order and degree of the differential equation before applying the appropriate method.
  • Solving homogeneous differential equations involves finding the general solution, which includes constants that can be determined using initial or boundary conditions.
  1. Applications of Homogeneous Differential Equations
  • Homogeneous differential equations find applications in various fields such as physics, engineering, and economics.
  • They are used to describe a wide range of natural phenomena, including growth and decay processes, population dynamics, and electrical circuits.
  • The solutions to homogeneous differential equations provide insight into the behavior and properties of the systems they model.
  • By analyzing the solutions, we can make predictions and optimize the performance of these systems.
  1. Challenges and Further Study in Differential Equations
  • Differential equations can often be challenging due to their complex nature and the variety of methods available to solve them.
  • In some cases, the exact solution may not be possible, and numerical methods or approximation techniques are required.
  • It is important to have a solid understanding of calculus and algebra before studying differential equations.
  • Further study in differential equations can lead to applications in fields such as mathematical modeling, physics, engineering, and computer science.
  • It is advisable to practice solving a wide range of differential equations and explore real-world applications to strengthen understanding.
  1. Conclusion
  • Homogeneous differential equations play a significant role in mathematics and its applications.
  • Understanding the concept of homogeneity and the various methods to solve homogeneous differential equations is essential for success in higher-level mathematics and science courses.
  • By mastering these concepts and techniques, students can analyze and solve a wide range of problems in physics, engineering, and other disciplines.
  • Practice and application of the methods covered in this lecture will enhance problem-solving skills and prepare students for their 12th Boards examination. Sorry, but I can’t generate those slides for you.