1. Definition and Terminology (ODE vs. PDE)

2. Order and Degree of a Differential Equation

3. General, Particular, and Singular Solutions

4. Initial Value Problems (IVP) and Boundary Value Problems (BVP)

5. Direction Fields and Slope Fields

Second-Order and Higher-Order ODEs 

1. Linear Second-Order ODEs with Constant Coefficients

2. Homogeneous and Nonhomogeneous Equations

3. Method of Undetermined Coefficients

4. Variation of Parameters

5. Reduction of Order

6. Mechanical Vibrations and Damped Harmonic Motion

Laplace Transforms 

1. Definition and Basic Properties

2. Inverse Laplace Transform

3. Laplace Transforms of Derivatives

4. Solving ODEs using Laplace Transforms

5. Step Functions and Dirac Delta Function

6. Convolution Theorem

Partial Differential Equations (PDEs) 

1. Classification of PDEs: Elliptic, Parabolic, Hyperbolic

2. Method of Separation of Variables

3. Fourier Series and Fourier Transforms

4. Heat Equation

5. Wave Equation

6. Laplace’s Equation

7. Boundary and Initial Conditions

Nonlinear Differential Equations 

1. Separable Differential Equations

2. Linear First-Order Equations

3. Exact Differential Equations

4. Integrating Factors

5. Homogeneous Equations

6. Bernoulli's Equation

7. Applications: Exponential Growth/Decay, Newton's Law of Cooling, Mixing Problems

Systems of Differential Equations 

1. Systems of First-Order Linear ODEs

2. Matrix Methods and Eigenvalue Problems

3. Phase Plane Analysis

4. Stability and Classification of Equilibrium Points

5. Applications: Predator-Prey Models, Competing Species

Series Solutions of Differential Equations 

1. Power Series Solutions

2. Frobenius Method

3. Special Functions (Bessel Functions, Legendre Polynomials)

Numerical Methods 

1. Euler's Method

2. Improved Euler (Heun’s) Method

3. Runge-Kutta Methods (RK2, RK4)

4. Finite Difference Methods for PDEs

Applications