Define hlambdax e-lambdax2 for any lambda gt 0 calculate

Question

Sample midterm 2 questions-

1. Consider - y''(x) - k²y(x) = f(x)

for -∞ < x < ∞, subject to the boundary conditions

lim_x→-∞ y(x) = lim_x→∞ y(x) = 0.

Find a Green function solution of the form

y(x) =  _-∞∫^∞G(x, x')f(x')dx'.

2. Define h_λ(x) = e^{-λx^2} for any λ > 0.

(a) Calculate the Fourier transform h^˜_λ of h_λ.

(b) Calculate the convolution g = h_λ ∗ h_µ.

(c) For the case of λ = 1 and µ = 2, sketch h_λ, h_µ, and g.

(d) Verify that the Fourier transform of g is equal to 2πh^˜_λh^˜_µ.

You may use the result that

_-∞∫^∞e^{-λx^2}dx = √(π/λ).

3. (a) By using Fourier and/or Laplace transforms, solve the partial differential equation

∂f/∂t + c(∂f/∂x) = b(∂²f/∂x²)

for the function f(x, t), where b > 0 and the initial condition is f(x, 0) = δ(x).

You may use the result that

-∞∫∞e^{-λx^2}dx = √(π/λ).

(b) Write down a Green function solution to the problem

∂f/∂t + c(∂f/∂x) = b(∂²f/∂x²)

subject to the initial condition f(x, 0) = g(x).

(c) Explicitly evaluate the solution in part (b) for the case when g(x) = e^-ax, and check that it satisfies the partial differential equation and initial condition.

4. (a) Calculate the Fourier series of

 (b) By using Parseval's theorem, show that

π²/8 = _n=0∑^∞ 1/(2n + 1)².

5. Use Laplace transforms to solve the differential equation

y'' + 6y' + 8y = e^-3t

subject to the boundary conditions y(0) = 1 and y'(0) = 0.