# Syllabus

### Introduction (Chapter 1)

• Equations and their solutions.
• Differential equations.
• ODE and PDE and their solutions.
• Order of a differential equation.

### First order ODEs (Chapter 2)

• Separable ODEs. Homogeneous Equations.
• Exact equations. Integrating factors.
• Linear equations. Reduction of order.

### Second order ODEs (Chapter 3)

• General solution. Use of a known solution to find another.
• Homogeneous equations with constant coefficients.
• The method of undetermined coefficients.
• The method of variation of parameters.

### Systems of first order ODEs (Chapter 10)

• Linear systems.
• Homogeneous linear systems with constant coefficients.
• Nonlinear systems. Volterra's prey-predator eauation.

### Nonlinear ODEs (Chapter 11)

• Autonomus systems. The phase plane.
• Types of critical points. Stability.
• Critical points and stability.
• Simple critical points of nonlinear systems

### Qualitative properties of solutions (Chapter 4)

• Oscilations and the Sturm separation theorem.
• The Sturm comparison theorem.

### Power series solutions (Chapter 5)

• Series solutions of first order ODEs.
• Second order linear ODEs.
• Regular singular points.
• Gauss's hypergemetric equation.