ETP 1.0 Differential Equations

1 Basic Concepts & Definitions

• • What is a differential equation?
• • Order and degree
• • Solutions: general and particular
• • Initial value problems (IVPs)

2 First-Order Form: y' = f(x, y)

• • Direction fields and solution curves
• • Existence and uniqueness
• • Classification of first-order ODEs

3 Separation of Variables

• • Separable equations
• • Solution techniques
• • Applications and examples

1 Homogeneous Differential Equations

• • Definition and recognition
• • Substitution method: y = vx
• • Separable solutions
• • Applications to modeling

2 Bernoulli's Equation

• • Standard form: y' + P(x)y = Q(x)yⁿ
• • Reduction to linear form
• • Solution technique via substitution

3 Problem-Solving Strategies

• • Identifying equation type
• • Choosing the right substitution
• • Step-by-step solution process

1 Exact Differential Equations

• • Definition: M(x,y)dx + N(x,y)dy = 0
• • Test for exactness: \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}
• • Finding potential functions
• • General solution method

2 Integrating Factors

• • When equations are not exact
• • Finding integrating factors \mu(x) or \mu(y)
• • Converting to exact form

3 Orthogonal Trajectories

• • Geometric interpretation
• • Finding orthogonal families of curves
• • Applications in physics and engineering

1 Second-Order Linear ODEs

• • Standard form: ay'' + by' + cy = 0
• • Characteristic equation method
• • Real, repeated, and complex roots
• • General solution structure

2 Constant Coefficients

• • Homogeneous equations
• • Solution based on characteristic roots
• • Wronskian and linear independence

3 Non-Homogeneous Equations

• • Particular solutions
• • Method of undetermined coefficients
• • General solution = complementary + particular

1 Method of Variation of Parameters

• • When undetermined coefficients fail
• • Assuming particular solution form
• • Determining parameter functions
• • General procedure and formula

2 Second-Order Applications

• • Finding particular solutions
• • Using Wronskian in computation
• • Complete solution construction

3 Higher-Order Extensions

• • Extending to higher-order ODEs
• • General patterns and formulas
• • Practical computation techniques

1 Cauchy-Euler Equation

• • Standard form: ax²y'' + bxy' + cy = 0
• • Solution by assuming y = x^m
• • Indicial equation and roots
• • Different cases: distinct, repeated, complex roots

2 Reduction of Order

• • When one solution is known
• • Finding second linearly independent solution
• • Applications to various equation types

3 Comprehensive Integration & Review

• • Unified framework: all ODE types
• • Decision tree for method selection
• • Exam preparation strategies
• • Final practice problems