Lecture Notes on Differential Equations

Differential Equations Lecture Notes

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  • Lecture Title: STQM2223 Differential Equations PM
  • Instructor: Dr. Azmin Sham Rambely
  • Date: 3/30/2023
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  • Start of Lecture: A prayer seeking knowledge.
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Introduction to Ordinary Differential Equations (ODE)
  • Objectives:
    • Introduce differential equations.
    • Discuss basic concepts in first-order ODEs.
    • Show methods of solving first-order ODEs.
    • Explain modeling examples involving first-order ODEs.
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Definition of Differential Equation
  • A differential equation consists of:
    • An unknown function (y(t))
    • A defining variable (t)
    • Derivatives of the unknown function (y’(t), y’’(t))
    • Constants.
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Definition of Ordinary Differential Equation (ODE)
  • A relationship between:
    • An independent variable
    • A dependent function and its derivatives.
  • Expressed as: dy/dt = f(t, y) where f is a given function.
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Classification of Differential Equations
  1. Ordinary Differential Equations (ODE):
    • Functions of a single variable (e.g., x or t).
  2. Partial Differential Equations (PDE):
    • Functions of multiple variables, involving partial derivatives.
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Order of Differential Equations
  • The order is the highest derivative present in the equation.
  • Example: d^n y/dx^n indicates it is n-th order.
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Linearity of Differential Equations
  • A linear ODE has derivatives that appear linearly, not as powers or products of y or its derivatives.
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Homogeneous vs Inhomogeneous Differential Equations
  • Homogeneous:
    • All terms (y and derivatives) set to zero.
  • Inhomogeneous:
    • Contains terms independent of y or its derivatives.
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Coefficients of Differential Equations
  • Constant Coefficients:
    • All coefficients are constants.
  • Variable Coefficients:
    • At least one coefficient is a variable.
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Solution of a Differential Equation
  • Definition: A solution of a n-th order ODE is a function differentiable n times satisfying
    F(x, y, dy/dx, ..., d^n y/dx^n) = 0.
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Types of Solutions
  • Explicit Solution: A solution expressed directly in terms of the independent variable.
  • Implicit Solution: Given by an equation that does not express y explicitly but satisfies the differential equation.
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Initial Value Problem (IVP)
  • A differential equation with specified values at a particular initial point.
  • To find a specific solution given these conditions.
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Boundary Value Problem (BVP)
  • Involves conditions at more than one point.
  • Different from IVP in that the conditions are not at a single point.
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Example: Initial Value Problem
  • Solve an initial value problem using separation of variables and integration.
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Setting Up Differential Equations
  • Example: Modeling dead leaves in a forest.
  • Define: D(t) = amount of leaves at time t.
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  • Consider the effects of decay and accumulation of leaves in the model:
    • Accumulation: 3 g/cm²/year.
    • Decay: 75% of leaves per year.
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Equilibrium Solution
  • At equilibrium, the rate of change is zero: dD/dt = 0.
  • Hence, D = 4 g/cm².
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Methods for Solving ODEs
  • Methods include:
    • Direct integration
    • Separable equations
    • Exact equations
    • Integrating factors
    • Variable changes.
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Example of Direct Integration
  • Solve differential equations of the form where f(x) is any function of x.
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Example of Separable Equations
  • Form: dy/dx = f(x)g(y) can be solved by separation and integration.
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Population Models
  • Changes in population relate to birth and death rates expressed through ODEs.
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Logistic Growth Model
  • A better approach for population growth, considering environmental constraints and the carrying capacity K.
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Conclusion
  • Recap key concepts on solving and applying differential equations to modeling.
  • End of Lecture Notes.