18.075 - Advanced Calculus for Engineers (Spring 2022)

Problem Sets

Problem Set 1 (due 02/10/2022, M)

• Prob 1. Find the general solution of the differential equation

  $y^{\prime\prime}-y=x^{2}+\sin3x+e^{4x}.$

  Determine the arbitrary constants in this solution if $y(x)$ satisfies the initial values $y(0)=0,$ $y^{\prime}(0)=1.$

• Prob 2. Prob 1a and 1c in Chapter 1.

Problem Set 2 (due 02/14/2022, Mon)

• Prob 1. Consider the following Sturm-Lioville problem:

  $y^{\prime\prime}=-\lambda y$, $0$ < $x$ < $2,$ $y(0)=y^{\prime}(2)=0.$

  1. Find the eigenvalues and the eigenfunctions of this problem.

  2. Let $y_{1}$ and $y_{2}$ be two eigenfunctions of this problem with eigenvalues $\lambda_{1}$ and $\lambda_{2}$ respectively. Show that the eigenfunctions are orthogonal to one another,.i.e.,

     $\int_{0}^{2}dx$ $y_{1}(x)y_{2}(x)=0,$ $\lambda_{1}\neq\lambda_{2}.$

• Prob 2. Problem 2 in chapter 2.

Problem Set 3 (due 02/22/2022, Tues)

• Prob 1.
  1. Is there an analytic function the imaginary part of which is $xe^{y}$?
  2. What is $a$ if $u=x^{4}+ax^{2}y^{2}+y^{4}$ is the real part of an analytic function? Find the imaginary part of this analytic function. Express this analytic function as a function of $z$.
• Prob 2. Let $u$ satisfies the two-dimensional Laplace equation $u_{xx}+u_{yy}=0$ in a region $R$ where $R$ is the disk $x^{2}+y^{2}$ < $9$ and $u$ satisfies the boundary condition $u=\cos^{2}4\theta$ at $r=3$. Find $u$.
• Prob 3.

  1. Prove the Liouville theorem that if an entire function $f(z)$ is bounded at infinity, it is a constant.

     Hint: show that the derivative of $f(z)$ is identically equal to zero.

  2. Prove that if an entire function divided by $z^{n}$ is bounded at infinity, it is a polynomial of order no more than $n.$

Problem Set 4 (due 03/02/2022 Wed)

• Prob 1. Consider the function $\dfrac{1}{z(9-z)}.$

  1. Find its Taylor series expanded about $z=1.$ Where is this series convergent?
  2. Find its various Laurent series expanded about $z=0$.

• Prob 2. Problem 4a and 4d in Chapter 2 of the text book.

Problem Set 5 (due 03/09/2022 Wed)

• Prob 1. Problem 4b and 4e in Chapter 2 of the text book.

Problem Set 6 (due 03/28/2022, Mon)

• Prob 1. Find the Fourier transform of $f(x)=e^{-x^{2}},-\infty$ < $x$ < $\infty.$

• Prob 2. Let

  $f(x)=e^{-2x},$     $x$ > $0,$

      $=0,$             $x$ < $0.$

  1. Find $\widetilde{f}(k)$, the Fourier transform of $f(x).$

  2. Apply the inversion formula

     $f(x)=\int_{-\infty}^{\infty}\dfrac{dk}{2\pi}e^{ikx}\widetilde{f}(k)$

     to determine $f(x)$ from the Fourier transform you found in (a).

  3. Set $k=0$ in the inversion formula to find $f(0).$ Explain the result you get.

• Prob 3. Let the solution of the differential equation

  $\dfrac{d^{2}x}{dt^{2}}-4x=\cos t,$ $t$ > $0$

  satisfy the initial conditions

  $x(0)=A,$ $\overset{\cdot}{x}(0)=B.$

  1. Find this solution with the use of Laplace transform.
  2. Find the general solution of the differential equation with the operator method you learned earlier in the semester. Determine the arbitrary constants in this solution with the initial conditions.
  3. Which of these two methods do you consider the more efficient? Why?

Problem Set 7 (due 04/20/2022, Wed)

• 
  

  Prob 1. Consider the equation

  $i\dfrac{\partial\Psi(x,t)}{\partial t}=-\dfrac{\partial^{2}\Psi (x.t)}{\partial x^{2}}+\rho(x,t),$

  where $\rho(x,t)$ is a given source function. The initial condition is

  $\Psi(x.0)=f(x),$

  and the boundary conditions are

  $\Psi(\pm\infty,t)=0.$

  1. Find the complementary solution of this problem.

  2. Find the partricular solution of this problem satisfying

     $\Psi_{p}(x.0)=0.$

     Express $\Psi_{p}(x.t)$ as an integral over the product of a Green's function and $\rho(x,t)$.

• 
  

  Prob 2. Let the Green's function $G(x-x^{\prime},t-t^{\prime})$ satisfies

  $(i\dfrac{\partial}{\partial t}+\dfrac{\partial^{2}}{\partial x^{2}})G(x-x^{\prime},t-t^{\prime})=\delta(x-x^{\prime})\delta(t-t^{\prime})$

  and the condition of causality

  $G(x-x^{\prime},t-t^{\prime})=0,$ $t$ < $t^{\prime}.$

  1. Find this Green function.
  2. Reproduce the solution of Prob 1 with the use of the partial differential equation in the presen problem.

• Prob 3. The wavefunction of an electron in the electrostatic field $eV(x,t)$ is

  $i\dfrac{\partial\Psi(x,t)}{\partial t}=[-\dfrac{\partial^{2}}{\partial x^{2}}+eV(x,t)]$ $\Psi(x.t).$

  Let the initial condition be

  $\Psi(x.0)=f(x).$

  Convert this problem into an integral equation. Indicate how you would solve this problem approximately if $e$ is small.

Problem Set 8 (due 05/04/2022, Wed)

• Prob 1. Let $f(\theta)=\theta,$ $0$ < $\theta$ < $\pi$.

  1. Expres this function in the Fourier cosine series

     $f(\theta)=a_{0}+\sum\limits_{n=1}^{\infty}A_{n}\cos n\theta$.

     Find $a_{0}$ and $A_{n}.$

  2. By setting $\theta$ to $0$ in this series, prove that

     $\dfrac{\pi^{2}}{8}=1+\dfrac{1}{3^{2}}+\dfrac{1}{5^{2}}+\cdot\cdot\cdot.$

  3. Use the formula

     $\dfrac{1}{\pi}\int_{0}^{\pi}f^{2}(\theta)d\theta=a_{0}^{2}+\dfrac{1}{2}\sum\limits_{n=1}^{\infty}A_{n}^{2}$.

     to prove that $\dfrac{\pi^{4}}{96}=1+\dfrac{1}{3^{4}}+\dfrac{1}{5^{4}}+\cdot\cdot\cdot.$

• Prob 2.Use the method of separation of variables to determine the solution of the Laplace equation

  $(\dfrac{\partial^{2}}{\partial x^{2}}+\dfrac{\partial^{2}}{\partial y^{2}})u=0$

  which holds in the annulus $1$ < $r$ < $2$ and satistfies the boundary conditions

  $u(1,\theta)=1+\cos\theta,$ $u(2,\theta)=2+\sin2\theta,$.

• Problem 3. Problem 4 in Chapter 5.