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
- Ordinary Differential Equations (ODE):
- Functions of a single variable (e.g., x or t).
- 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.