Advance Maths: Part I: Linear Differential Equations

Delve into the world of Linear Differential Equations (LDE) with this comprehensive course! Master the fundamentals and advanced techniques for solving a wide range of LDEs, crucial for various scientific and engineering disciplines.

What You'll Learn:

• Second and Higher Order LDEs: Gain expertise in solving LDEs of second order and beyond, specifically those with constant coefficients.
• Homogeneous Equations: Understand and apply methods for solving homogeneous linear differential equations.
• Variation of Parameters: Become proficient in the powerful variation of parameters technique for finding particular solutions.
• Euler's/Cauchy's Equations: Learn to solve Euler's equations and related Cauchy's equations.
• Legendre's Form: Explore and utilize Legendre's form for solving specific LDEs.
• Simultaneous LDEs: Develop skills in solving systems of simultaneous linear differential equations, including symmetrical equations.
• Applications of LDEs: Discover real-world applications of LDEs in diverse fields.

This course provides a thorough exploration of LDEs, complete with numerous solved examples and exercises to facilitate self-assessment and reinforce your understanding. Prepare thoroughly for exams and build a strong foundation in this essential topic.

Key Topics Covered:

• Exact Differential Equations: Learn to identify and solve exact differential equations and equations that can be reduced to an exact form.
• Equations Reducible to Linear Form: Master techniques for transforming non-linear differential equations into a linear form.
• Bernoulli’s Equation: Understand and solve Bernoulli's equation.
• Applications to Orthogonal Trajectories: Explore how differential equations are used to determine orthogonal trajectories.
• Newton's Law of Cooling: Apply differential equations to model Newton's Law of Cooling.
• Kirchhoff's Law of Electrical Circuits: Learn how to use differential equations to analyze Kirchhoff's Law of Electrical Circuits.
• Rectilinear Motion: Model and solve problems involving rectilinear motion.
• Simple Harmonic Motion: Understand and apply the principles of simple harmonic motion.
• One-Dimensional Conduction of Heat: Utilize differential equations to analyze one-dimensional heat conduction.

Detailed Breakdown:

1. First-Order Differential Equations: A foundational understanding of first-order differential equations.
2. Linear Differential Equations with Constant Coefficients: A deep dive into LDEs of nth order with constant coefficients. This includes:
   • Method of Variation of Parameters
   • Cauchy’s and Legendre’s Differential Equations
   • Simultaneous & Symmetric Differential Equations
   • Modeling: Practical applications in bending of beams, whirling of shafts, and mass-spring systems.
3. Complementary Function (C.F.) & Particular Integral (P.I.): Learn to define and find the complementary function (C.F.) and particular integral (P.I.) of LDEs.
4. Cauchy's and Legendre's Homogeneous Linear Differential Equations: Master techniques for solving Cauchy's homogeneous linear differential equation and Legendre's homogeneous equation.
5. Modeling of Mass-Spring Systems: Further explore the modeling of mass-spring systems, including free and forced damped and undamped systems.
6. Introduction to Vibration: Gain an introduction to undamped and damped vibration.