Series solutions of differential equations
• legendre, bessel, hermite, and laguerre functions
• Legendre’s equation
• Leibniz’ rule for differentiating products
• Rodrigues’ formula
• Generating function for legendre polynomials
• Complete sets of orthogonal functions
• Orthogonality of the legendre polynomials
• Normalization of the legendre polynomials
• Legendre series
• The associated legendre functions
• Generalized power series or the method of frobenius
• Bessel’s equation
• The second solution of bessel’s equation
• Graphs and zeros of bessel functions
• Recursion relations
• Differential equations with bessel function solutions
• Other kinds of bessel functions
• The lengthening pendulum
• Orthogonality of bessel functions
• Approximate formulas for bessel functions
• Series solutions; fuchs’s theorem
• Hermite functions; laguerre functions; ladder operators 18
Partial differential equations
• Laplace’s equation; steady-state temperature in a rectangular plate
• The diffusion or heat flow equation; the schr?dinger equation
• The wave equation; the vibrating string
• Steady-state temperature in a cylinder
• Vibration of a circular membrane
• Steady-state temperature in a sphere
• Poisson’s equation
• Integral transform solutions of partial differential equations 18
Calculus of variations
• The euler equation
• Using the euler equation
• The brachistochrone problem; cycloids
• Several dependent variables; lagrange’s equations
• Isoperimetric problems
• Variational notation