Use separation of variables to find a series solution of utt = c 2uxx subject to...

Use separation of variables to find a series solution of u_tt = c ²u_xx subject to u(0, t) = 0,

u_x(l, t) + u(l, t) = 0, u(x, 0) = φ(x), & u_t(x, 0) = ψ(x) over the domain 0 < x < `, t > 0. Provide an equation that identifies the eigenvalues and sketch a graph depicting this equation. Clearly identify the eigenfunctions

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Colby Messinger answered 2 years ago

(PDE) Write the soln using separation of variables , in the form of fourir series: Utt=Uxx...

(PDE) Write the soln using separation of variables , in the form of fourir series: Utt=Uxx boundary: U(t,0)=0=U(t,pi) initial : initial: U(0,x)=1 and Ut(0,x)=0

Problem #2: Use separation of variables with λ = 36 to find a product solution to...

Problem #2: Use separation of variables with λ = 36 to find a product solution to the following partial differential equation, y ∂2u ∂x2 + ∂u ∂y = 0 that also satisfies the conditions u(0, 0) = 9 and ux(0, 0) = 4.

Using the method of separation of variables and Fourier series, solve the following heat conduction problem...

Using the method of separation of variables and Fourier series, solve the following heat conduction problem in a rod. ∂u/∂t =∂2u/∂x2 , u(0, t) = 0, u(π, t) = 3π, u(x, 0) = 0

Obtain the general solution of the following equations: r Utt − c^2 r Urr − 2...

Obtain the general solution of the following equations: r Utt − c^2 r Urr − 2 c^2 Ur = 0, c = constant,

Problem #1 : Using separation of variables and the formalism demonstrated in class, find the general...

Problem #1 : Using separation of variables and the formalism demonstrated in class, find the general solution to the Helmholtz equation (also known as the time independent wave equation) : ∇^2 F + k^2 F = 0 Assume that k is a real number. For clarification, the general solution is the solution in the case where boundary conditions are not specified. Problem #2 : Express the following functions as an infinite Fourier sine series using sin(nπx/a) a)f(x) = x b)f(x)...

Determine the solution of the following initial boundary-value problem using the method of separation of Variables...

Determine the solution of the following initial boundary-value problem using the method of separation of Variables Uxx=4Utt 0<x<Pi t>0 U(x,0)=sinx 0<=x<Pi Ut(x,0)=x 0<=x<Pi U(0,t)=0 t>=0 U(pi,t)=0 t>=0

Find a series solution ofy′′−xy′+ 2y= 0.

Use the separation of variables method to solve the following problem. Consider a long, narrow tube...

Use the separation of variables method to solve the following problem. Consider a long, narrow tube connecting two large, well-mixed reservoirs containing a small concentration of N2 in another inert gas. The tube length is L = 100 cm. To establish an initial concentration profile in the tube, each reservoir is held at a fixed concentration: Reservoir 1 contains no N2 and reservoir 2 has 2 × 10−6 mol/cm3 of N2. (a) At t = 0, the concentrations of the...

Separation of Variables: The rate at which fuel is being used by a Wartsila-Sulzer RTA96-C Turbocharged...

Separation of Variables: The rate at which fuel is being used by a Wartsila-Sulzer RTA96-C Turbocharged Diesel Engine is proportional to the amount of fuel already consumed at time t. If there are 2 gallons consumed when t 0 and 3 gallons consumed when t 5, how many gallons will be used when t = 9?

Solve the following wave equation using Fourier Series a2uxx = utt, 0 < x < 1,...

Solve the following wave equation using Fourier Series a2uxx = utt, 0 < x < 1, t > 0, u(0,t) = 0 = u(1,t), u(x,0) = x2, ut(x,0) = 0

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