IN THIS CHAPTER we use the series discussed in Chapter 11 to solve partial differential equations that arise in problems of mathematical physics.
SECTION 12.1 deals with the partial differential equation
\[
u_t=a^2u_{xx},
\]
which arises in problems of conduction of heat.
SECTION 12.2 deals with the partial differential equation
\[
u_{tt}=a^2u_{xx},
\]
which arises in the problem of the vibrating string.
SECTION 12.3 deals with the partial differential equation
\[
u_{xx}+u_{yy}=0,
\]
which arises in steady state problems of heat conduction and potential theory.
SECTION 12.4 deals with the partial differential equation
\[
u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0,
\]
which is the equivalent to the equation studied in Section 1.3 when the independent variables are polar coordinates.
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