This chapter aims to demonstrate how boundary value problems in mathematical physics can be solved using methods from previous chapters. It does so by solving various specific problems that illustrate the main types discussed in Chapter 7. Additional applications are covered in the exercises. The primary solution method is Fourier's separation of variables, along with the Sturm–Liouville theory from Chapter 8. Chapter 9 is divided into seven sections. Section 9.1 reviews one-dimensional heat diffusion problems (heat equation $ u_t = u_{xx} $) discussed in Chapter 8. Section 9.2 addresses boundary value problems for vibrating strings (wave equation $ u_{tt} = c^2 u_{xx} $). Section 9.3 applies Fourier's method to steady-state temperatures in plates (two-dimensional Laplace equation $ u_{xx} + u_{yy} = 0 $). Section 9.4 deals with transient temperatures in plates (heat equation $ u_t = u_{xx} + u_{yy} $). Section 9.5 analyzes vibrations of circular drums (wave equation $ u_{tt} = c^2 (u_{xx} + u_{yy}) $). Section 9.6 examines steady-state heat diffusion in solids (three-dimensional Laplace equation $ u_{xx} + u_{yy} + u_{zz} = 0 $). Finally, Section 9.7 presents an alternative method based on the Laplace transform. Section 9.1 focuses on transient heat diffusion problems where temperature is a function of time and one spatial coordinate. Eight such problems are solved.This chapter aims to demonstrate how boundary value problems in mathematical physics can be solved using methods from previous chapters. It does so by solving various specific problems that illustrate the main types discussed in Chapter 7. Additional applications are covered in the exercises. The primary solution method is Fourier's separation of variables, along with the Sturm–Liouville theory from Chapter 8. Chapter 9 is divided into seven sections. Section 9.1 reviews one-dimensional heat diffusion problems (heat equation $ u_t = u_{xx} $) discussed in Chapter 8. Section 9.2 addresses boundary value problems for vibrating strings (wave equation $ u_{tt} = c^2 u_{xx} $). Section 9.3 applies Fourier's method to steady-state temperatures in plates (two-dimensional Laplace equation $ u_{xx} + u_{yy} = 0 $). Section 9.4 deals with transient temperatures in plates (heat equation $ u_t = u_{xx} + u_{yy} $). Section 9.5 analyzes vibrations of circular drums (wave equation $ u_{tt} = c^2 (u_{xx} + u_{yy}) $). Section 9.6 examines steady-state heat diffusion in solids (three-dimensional Laplace equation $ u_{xx} + u_{yy} + u_{zz} = 0 $). Finally, Section 9.7 presents an alternative method based on the Laplace transform. Section 9.1 focuses on transient heat diffusion problems where temperature is a function of time and one spatial coordinate. Eight such problems are solved.