Solve the eigenvalue problem Lf (x) = Af(x), where k df(x) i dx In the region -a < x< a, subject to the boundary condition that f (a) = f(-a) Lf(x) = – ßxf (x) -

Q: Consider the ODE t²y" + 6ty' + 6y = e4", for t > 0. (a) Find the homogeneous solution of the ODE.…

A: See the attachment

Q: Use the Laplace transform to solve the initial-value problem y" – y - 6y = 0, y(0) = 1, y(0) = 2.…

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Q: Use laplace transform to solve the initial value problem; a) y'+y=alpha(t-1),y(0)=2

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Q: (d) Use Fourier Sine transform to solve: a'u du əx² ay² 00; [u(0,y)=0, u(x,0)=f(x); (u(x,y) →0, as…

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Q: + x + + 3y where x(0) = 0 and y(0) = 0. ||||

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Q: Solve the ODE: φ'' = - λφ for 0 < x < π , subject to the boundary conditions φ(0) = 0 φ' (π) =…

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Q: Use the eigenfunction solution of the regular Sturm-Liouville problem: dx +ây = 0, y'(0) = y(1) = 0…

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Q: 2. Eigenvalue Problem with first derivative in linear operator. Find the eigenvalues and…

A: The given boundary value problem,

Q: Integrate (a) S2(* dt (b) S(x+1)³ dx (c) cos(-x)(x²)dx

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Q: 10. Solve the initial value problem x(t + 1)- ) xt) + () ()- x(t) + x(0) = %3D by Z-transform.

A: For the equation xt+1=1122xt+10 taking z transform of LHS and RHS:…

Q: Suppose U solves the heat equation on the real lineUt = 4Uxx, x ∈ Rwith initial valueU(x, 0) = (4, x…

A: The objective of the question is to solve the heat equation using the Fourier-Poisson formula and…

Q: Determine the value of the Wronskian W(y1, y2) of y" - 2y + 1 = e2*(x + 1) if x-0.2 by using…

A: Given, y''-2y'+1=e2xx+1. The objective of the problem is to obtain the value of the Wronskian if…

Q: A2 Use the Fourier Transform Method to find the solution u(x,t) of the initial value problem: ut +…

A: Solve for u(x,t): We now have:U(ω, t) = F(ω) * e^(-iωt^2/2)Invert the Fourier Transform to obtain…

Q: (b) Find the Wronskian W[y1,Y2](t) of y1 and y2-

A: W[y1,y2]= -t2et

Q: Solve the linear first-order ODE y' + In(x) y = xe3, a> 0, where the prime stands for differei with…

A: Given: y'+ln(x)y=x-xe3x

Q: Show that the integral equation TT g(x) = 1 (sin x sin 2t) g(t) dt has no eigenvalue

A: Solution:-

Q: Use Laplace Transforms to solve the initial value problem where [y'-4 y=f(x), y(0) = 0] f(x) = 2 3…

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Q: If g(x) = -S^*[e²− ln(2t+1)]dt, find g''(x) . dx 2 (A (2x + 1)2 Be* -In(2x+1) 4 (C) Ce¹+ (2x + 1)² 4…

A: Here is the solution of the given problem.  By using derives under Integral.

Q: Solve the initial value problem using Laplace transform b) y" + 3y' + 2y = 2 8(t-3), y(0)=0,…

A: Laplace and inverse Laplace transform

Q: Solve PDE 1 = e=*t u. u̟ =e¯*t Xx k (o0) u(0,t) =0 u(L,t) = 0 u(x,0) = L -x?

A: The given PDE is,uxx-1kut=e-xt

Q: f(x) = | e 'cos(af(s))ds, 0<x< 1, 0 < a < 1, -SX

A: Given: f(x)=∫01e-sxcosαf(s)ds, 0≤x≤1, 0&lt;α&lt;1

Q: d If g(x) = - S* [e'− ln(2t+1)]dt, find g''(x) dx -2 2 A ex. (2x + 1)² 4 (В B et + (2x + 1)² Ⓒe*…

A: Given that gx=ddx∫-2xet-ln2t+1dt

Q: solve the initial value problem with Kernel function   y’’ +6y’= g(t) where y(0)=1 , y’(0)=0

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Q: Solve the IVP dx dt [ x(t) = 12 0 X, -24 0 x(0) = [13] 181

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Q: Consider the IVP dy = (). Iy(20 -- y), y(0) = 1. dt %3D (a) Solve the IVP, finding y explicitly as a…

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Q: 12:06 ■I 5G (8) If f(z) = √1+dt and g(y) = f (x)dx then find g"(#/3).

A: Thank you

Q: Let y₁(x) and y₂(x) be any two twice differentiable functions, not necessarily eigenfunctions, in an…

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Q: For the linear integral equation 1/2 (x) = x + √(5) de, +To(5) 0 d5, \-(214) the resolvent kernel…

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Q: Write the function f(x) = x(1-x) as an eigenfunction expansion of the eigenfunctions that correspond…

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Q: Q5) (a) Solve by using Laplace transform the partial differential equation 8², =9. y(0,t)= 0, y(2,t)…

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