4.6.5 Helmholtz Equation And Integral Representation -- 4.7 Lommel Differential Equation -- 4.8 Hankel Function -- 4.9 Kelvin Functions -- 4.10 Legendre Equation -- 4.10.1 Series Solution -- 4.10.2 Rodrigues Formula -- 4.11 Associated Legendre Equation -- 4.12 Hypergeometric Function -- 4.12.1 Frobenius Series Solution -- 4.12.2 Confluent Hypergeometric Function -- 4.12.3 Kummer's Classification Of Hypergeometric Series -- 4.12.4 Hypergeometric Series Versus Other Functions -- 4.13 Generalized Hypergeometric Equation -- 4.14 Movable Singularities And Painleve Equations -- 4.15 Summary And Further Reading -- 4.16 Problems -- Chapter 5. System Of First Order Differential Equations -- 5.1 Introduction -- 5.2 Reduction Of N-th Order Ode To System Of Equations -- 5.3 Analytic Solution By Elimination -- 5.4 Determinacy Of System Of Equations -- 5.5 Review On Matrix -- 5.5.1 Hermitian Matrix -- 5.5.2 Eigenvalue Problem -- 5.5.3 Differentiation Of Matrix -- 5.6 Homogeneous System With Constant Coefficients -- 5.6.1 Case 1: Distinct Eigenvalues -- 5.6.2 Case 2: Repeated Eigenvalues -- 5.6.3 Case 3: Complex Eigenvalues -- 5.7 Nonhomogeneous System With Constant Coefficient -- 5.7.1 Undetermined Coefficients -- 5.7.2 Variation Of Parameters -- 5.8 System Of Nonlinear Ode -- 5.9 Stability Of Autonomous System -- 5.10 Summary And Further Reading -- 5.11 Problems -- Chapter 6. First Order Partial Differential Equations (PDEs) -- 6.1 Introduction -- 6.2 First Order Pde -- 6.3 Hyperbolic Equation -- 6.3.1 Transport With Decay -- 6.3.2 Non-uniform Transport -- 6.3.3 Nonlinear Transport And Shock Waves -- 6.4 Airy's Method For Hyperbolic Equation -- 6.5 Geometric Interpretation Of Linear Pde -- 6.6 Cauchy-kovalevskaya Theorem