(∂^2 u)/(∂t^2) =9 (∂^2 u)/(∂x^2),0 0 u(0, t) =u(π, t)=0, t>0 u(x, 0) =sin4 x+7 sin5 x,

step 1 Okay, we start off with U of X comma T, which is equal to the summation of n equal and infinity

(∂^2 u)/(∂t^2) =9 (∂^2 u)/(∂x^2),0<x<π, t>0 u(0, t) =u(π,...

$$ \frac{\partial^{2} u}{\partial t^{2}} $$$=9 \frac{\partial^{2} u}{\partial x^{2}}$,$0<$x$<\pi$, $t>0$ $$ u(0, t) $$$=u(\pi, t)=0, \quad t>0$ $$ u(x, 0) $$$=\sin 4 x+7 \sin 5 x$, $0<$x$<\pi$, $$ \frac{\partial u}{\partial t}(x, 0) $$$(x, 0)=$ $x, \quad 0<$x$<\pi / 2$ $\pi-x, \quad \pi / 2<$x$<\pi$

$$
\frac{\partial^{2} u}{\partial t^{2}}
$$$=9 \frac{\partial^{2} u}{\partial x^{2}}$,$0<$x$<\pi$, $t>0$
$$
u(0, t)
$$$=u(\pi, t)=0, \quad t>0$
$$
u(x, 0)
$$$=\sin 4 x+7 \sin 5 x$, $0<$x$<\pi$,
$$
\frac{\partial u}{\partial t}(x, 0)
$$$(x, 0)=$ $x, \quad 0<$x$<\pi / 2$
$\pi-x, \quad \pi / 2<$x$<\pi$

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Key Concepts

Wave Equation

The wave equation is a second order partial differential equation that models the propagation of waves through a medium, emphasizing the relationship between spatial and temporal variations. It typically involves a constant factor (squared wave speed) multiplying the spatial derivative, and is used in contexts such as vibrations, acoustics, and electromagnetics.

Separation of Variables

Separation of Variables is a technique used to solve PDEs by assuming that the solution can be written as a product of functions, each dependent on a single independent variable. This method transforms a complex PDE into simpler ordinary differential equations, which can be solved using standard methods.

Fourier Series

Fourier Series is a tool for representing functions as sums of sine and cosine functions. In the context of solving PDEs, it is used to expand and match initial or boundary conditions, particularly when these conditions are specified on a finite interval and the eigenfunctions are orthogonal.

Sturm-Liouville Theory

Sturm-Liouville Theory underpins the process of finding eigenfunctions and eigenvalues when solving boundary value problems associated with differential equations. This theory ensures that the eigenfunctions form an orthogonal basis, which facilitates the use of Fourier series in representing solutions to PDEs.

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