This series has been completed on 21 July, 2026. Click on the drop-down lists to see all the posts.
Ch 1: First-order ODEs
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- ODEs 1-1: Three Classifications of ODEs
- ODEs 1-2: Separable First-order ODEs
- ODEs 1-3: First-order Linear ODEs
- ODEs 1-4: More First-order ODEs Examples
- ODEs 1-5: Exact First-order ODEs (Part 1)
- ODEs 1-6: Exact First-order ODEs (Part 2)
- ODEs 1-7: Identification of Special Differentials
- ODEs 1-8: Homogeneous (of Degree 0) First-order ODEs
- ODEs 1-9: Isobaric ODEs
- ODEs 1-10: Bernoulli ODEs
- ODEs 1-11: Riccati ODEs
- ODEs 1-12: Higher-degree First-order ODEs (Part 1)
- ODEs 1-13: Higher-degree First-order ODEs (Part 2)
- ODEs 1-14: Orthogonal Trajectories
- ODEs 1-15: Existence and Uniqueness of Solution for First-order ODEs
Ch 2: Second-order ODEs
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- ODEs 2-1: Some Basics about Second-order ODEs
- ODEs 2-2: Second-order Constant-coefficient Homogeneous ODEs (Part 1)
- ODEs 2-3: Second-order Constant-coefficient Homogeneous ODEs (Part 2)
- ODEs 2-4: Second-order Constant-coefficient Homogeneous ODEs (Part 3)
- ODEs 2-5: Damped Spring-Mass System, Harmonic Oscillators
- ODEs 2-6: Method of Undetermined Coefficients (Part 1)
- ODEs 2-7: Method of Undetermined Coefficients (Part 2)
- ODEs 2-8: Variation of Parameters
- ODEs 2-9: Forced Harmonic Oscillators, Resonance
- ODEs 2-10: Euler Form
- ODEs 2-11: Canonical Form
- ODEs 2-12: Reduction of Order (Part 1)
- ODEs 2-13: Reduction of Order (Part 2)
- ODEs 2-14: Exact ODEs with Higher-order Derivatives
- ODEs 2-15: Higher-order Homogeneous/Isobaric Equations
Ch 3: Series Solution
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- ODEs 3-1: Prototype of Series Solution
- ODEs 3-2: Series Solution for First-order ODEs
- ODEs 3-3: Series Solution for Second-order ODEs, Ordinary Points (Part 1)
- ODEs 3-4: Series Solution for Second-order ODEs, Ordinary Points (Part 2)
- ODEs 3-5: Oscillation, Airy Equation/Functions (Part 1)
- ODEs 3-6: Oscillation, Airy Equation/Functions (Part 2)
- ODEs 3-7: Regular Singular Points, Frobenius Solution (Part 1)
- ODEs 3-8: Regular Singular Points, Frobenius Solution (Part 2)
- ODEs 3-9: Hypergeometric Equation/Function (Part 1)
- ODEs 3-10: Hypergeometric Equation/Function (Part 2)
- ODEs 3-11: Point at Infinity, Canonicity of Hypergeometric Equation
- ODEs 3-12: Confluent Hypergeometric Equation
Ch 4: Laplace Transform
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Ch 5: Sturm-Liouville Theory
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- ODEs 5-1: Sturm-Liouville Equation
- ODEs 5-2: Hermitian Operators
- ODEs 5-3: Spectral Theorem, Fourier Series
- ODEs 5-4: Bessel’s Inequality, \(L^2\) Convergence
- ODEs 5-5: Termwise Differentiation and Integration of Fourier Series
- ODEs 5-6: Revisiting Green’s Function (Part 1)
- ODEs 5-7: Revisiting Green’s Function (Part 2)
Ch 6: Special Functions
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- ODEs 6-1: Legendre Equation/Polynomials
- ODEs 6-2: Legendre Series
- ODEs 6-3: Gauss-Legendre Quadrature Integration
- ODEs 6-4: Chebyshev Equation/Polynomials
- ODEs 6-5: Minimax Properties of Chebyshev Nodes
- ODEs 6-6: Hermite Equation/Polynomials
- ODEs 6-7: Hermite Polynomials for Equatorial Waves
- ODEs 6-8: Laguerre Equation/Polynomials
- ODEs 6-9: Bessel Equation/Functions (Part 1)
- ODEs 6-10: Bessel Equation/Functions (Part 2)
- ODEs 6-11: Modified Bessel Equation/Functions
Ch 7: Introduction to PDEs
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- ODEs 7-1 (PDEs): Three Major Types of PDEs, Boundary Conditions
- ODEs 7-2 (PDEs): Heat Equation (Part 1)
- ODEs 7-3 (PDEs): Heat Equation (Part 2)
- ODEs 7-4 (PDEs): Heat Equation (Part 3)
- ODEs 7-5 (PDEs): Laplace Equation in Rectangular Coordinates
- ODEs 7-6 (PDEs): Laplace Equation in Polar Coordinates
- ODEs 7-7 (PDEs): Laplace Equation in Cylindrical Coordinates
- ODEs 7-8 (PDEs): Laplace Equation in Spherical Coordinates
- ODEs 7-9 (PDEs): Wave Equation (Part 1)
- ODEs 7-10 (PDEs): Wave Equation (Part 2)
- ODEs 7-11 (PDEs): Uniqueness Theorems for Second-order Linear PDEs
