Since ends are insulated no heat can flow through ends of bar. Therefore u_x(0,t) = 0 = u_x(l,t).

Boundary conditions are always used first to solve partial differential equations. Using se boundary conditions, we can remove one constant thus making only one constant remaining to remove. The last constant is removed using initial conditions.

\(u(x,t) = X(x) T(t) \)
\(x^3 X’Y+y^2 Y’X=0\)
\(\frac{X?}{ X} = \frac{k}{x^3} \) which implies \(X = ce^{\frac{k}{2x^2}}\)
\(\frac{Y?}{Y} = \frac{-k}{y^2} \) which implies \(Y = c? e^{\frac{k}{y}}\)
\(u(x,t) = cc?e^{\frac{k}{2x^2}} e^{\frac{k}{y}} \)
\(u(0,y) = 10e^{\frac{5}{y}}= cc?e^{\frac{k}{y}} \)
Therefore k = 5 and cc? = 10
Hence, \(u(x,t) = 10e^{\frac{-5}{2x^2}} e^{\frac{5}{y}}. \)

Anor definition of initial condition may be stated as, ?any of a set of starting-point values belonging to or imposed upon variables in an equation that has one or more arbitrary constants?

u(x,t) = (c cospx + c? sinpx) (c?? e^-p2t)
When x=0, c=0 and when x=1, p=n?.
When t=0, 3 sin (n?x) = \(?_{n=1}^? \) b_n e^{-n2 ?2 t} sin?(n?x)
Therefore b_n =3 for all n
Hence solution is \(3?_{n=1}^? \) e^{-n2 ?2 t} sin?(n?x).

D’Alembert’s formula for obtaining solutions to wave equation is named after him. The wave equation is sometimes referred to as d’Alembert’s equation.

Separation of variables (or Fourier method) is a method of solving ordinary and partial differential equations, in which we can rewrite an equation so that each of two variables occur on different sides of equation.

Since given problem is 1-Dimensional wave equation, solution should be periodic in nature. If k is a positive number, n solution comes out to be (c₇ e^px?c+e^-px?cc₈)(c₇ e^pt+e^-ptc₈) and if k is positive solution comes out to be (ccos(px/c) + c?sin(px/c))(c??cospt + c???sinpt). Now, since it should be periodic, solution is (ccos(px/c) + c?sin(px/c))(c??cospt + c???sinpt).

The order of an equation is defined as highest derivative present in equation. Hence, in given equation, \(\frac{?^2 z}{?x^2}-(\frac{?z}{?y})^5+\frac{?^2 z}{?x?y}=0\), order is 2.

The wave equation arises in fields like acoustics, electromagnetics, and fluid dynamics. Historically, problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d’Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange. In 1746, d?Alembert discovered one-dimensional wave equation, and within ten years Euler discovered three-dimensional wave equation.