Exam Reviews

Review Sheets for class exams and the final exam will be posted on this page at least one week prior to the exam.

The review sheet for the Final Exam can be found here. The answer key and formula sheet appear at the end of the document.

The review sheet for Exam #1 can be found here. The answer key appears at the end of the document.

The review sheet for Exam #2 can be found here. The answer key appears at the end of the document.

The review sheet for Exam #3 can be found here. The answer key appears at the end of the document.

2017 REVIEW SHEETS

Below are the review sheets from the last time I taught this course, Spring 2017, for your reference.

The review sheet for the Final Exam can be found here. The answer key and formula sheet appear at the end of the document.

The review sheet for Exam #3 can be found here. The answer key appears at the end of the document.

The review sheet for Exam #2 can be found here. The answer key appears at the end of the document.

The review sheet for Exam #1 can be found here. The answer key appears on the second page of the document.

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Solution of Linear Differential Equation in specific form.
I solved the standard format, arrived in equation with integrating factor. Followed partial derivation but couldn't move past. Help is needed. Question : https://imgur.com/h89f7hu Can anyone edit it here, not aware about that.
Solution of $V'(t)=AV(t)+V(t)A^T+\sigma^2I_m$, where $A$ is the discretized Neumann-Laplacian
I'm considering a PDE involving the Laplacian on $[0,1)^2$. I'm discretizing the problem using the finite difference approach. The resolution discretized Laplcian opertor $A$ with Neumann boundary conditions is of the form as in the picture below: with $h=1$. For this I need to solve the equation $$V'(t)=AV(t)+V(t)A^T+\sigma^2I_m$$ with $V(0)=0$, where $\sigma>0$. Do I need […]
Finding the roots of $P_1-P_2$, given those of $P_1+P_2$
Given two polynomials $P_1(x)$ and $P_2(x)$ and the roots of the polynomial $Q_1(x) = P_1(x)+P_2(x)$, which are al real. I would like to find the roots of the polynomial $Q_2(x) = P_1(x)-P_2(x)$, which I assume will also be real. My strategy was given a root $r$ of $Q_1(x)$ I can try to deform $Q_1(x)$ into […]
A solution procedure by "Fourier-Laplace transform" to Airy function (from oscillatory to exponential behavior)?
In studying the Airy equations (with final objective is to find the asymptotic behavior of the oscillatory solution to growing (frequency $\omega\to\infty$) decaying ($\omega\to0$) exponential behavior). Part of motivation is also due to the attempts to answer this interesting post Oscillatory and exponential behaviour of second order ODE. The material I was reading is this […]
"Heteroclinic orbit" for non-autonomous system?
Apologies if I am using language incorrectly, but I want to consider a non-autonomous ODE for $u(x),v(x)$ that is approximately autonomous in the limits of both $x\rightarrow\pm \infty$. An example of the kind of ODE I am dealing with is, $$u'=v,\quad v'= (u^2+e^x u)^2-u$$ When $x\rightarrow -\infty$, we can ignore the term with the exponential […]
Solving a 2nd order ODE. [closed]
Hi I'm trying to solve this ODE, It is part of a damped wave equation PDE where I am now trying to solve the $T$ equation. $$T'' + \gamma T' + T (n\pi)^2 = 0$$ I have the aux equation which is $$r^2 + \gamma r + (n\pi)^2 = 0$$ And I have obtained the […]
Some questions about existence and uniqueness theorem
The solution of the ODE $$y'-\frac{2}{x}y=0$$ is $$y(x)=cx^2$$ for some $c\in\mathbb{R}$. If $y(0)=0$, then $$y(x)=0,\quad y(x)=x^2$$ are both solutions to the IVP. Why this doesn't contradict the existence and uniqueness theorem? And since $$y'=\frac{2}{x}y$$ then $$f(x,y)=\frac{2y}{x}\Rightarrow \frac{\partial f}{\partial y}=\frac{2}{x}$$ i.e., $f_{y}(x,y)$ is not continuous at $x=0$. So, how can I still find a solution at […]
How to get/determine the Green's Function?
I want to determine the Green's function of $$t y^{\prime \prime}(t)+y^{\prime}(t)=t, \quad y(1)=y(e)=0.$$ I have even solved the ODE with the initial conditions, but I do not know how to determine the Green's function: We have a second-order linear ODE. Let $$\frac{dy(t)}{dt} = v(t),$$ which gives $$\frac{d^{2}y(t)}{dt^{2}} = \frac{dv(t)}{dt}.$$ Now, substitute $1 = \frac{d}{dt}(t):$ $$t […]
Power series solution to $x^2y''+y'+y=0$ around $x=0$
If an ODE of the form $$y''+p(x)y'+q(x)y=0$$ has $p(x)$ and $q(x)$ that are analytic at $x_0$ then we can suppose a power series solution as such: $$y(x)=\sum_{n\geq 0} a_n(x-x_0)^n$$ If instead they have a first order pole at $x_0$ we have to suppose this other one: $$y(x)=\sum_{n\geq 0} a_n(x-x_0)^{n+r}$$ But what about a power series […]
Contradicting solutions in the Fourier space
Consider the following differential equations \begin{equation*} \partial_t X(t) = -i \alpha X(t) + \beta Y(t), \quad \partial_t Y(t) = -i \alpha Y(t) - \beta X(t).~~~~~ (1) \end{equation*} with $\alpha$, $\beta$ some complex constants (independent of $t$). $\textbf{Case-1 [Time domain]:}$ The solution (with initial condition, say $X(0) = 1, Y(0) = 0$) reads \begin{equation*} X(t) = […]

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The OpenLab at City Tech:A place to learn, work, and share

The OpenLab is an open-source, digital platform designed to support teaching and learning at City Tech (New York City College of Technology), and to promote student and faculty engagement in the intellectual and social life of the college community.

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The OpenLab at City Tech:A place to learn, work, and share

The OpenLab is an open-source, digital platform designed to support teaching and learning at City Tech (New York City College of Technology), and to promote student and faculty engagement in the intellectual and social life of the college community.