Hi everyone,

The exam #3 review sheet is posted under “Classroom Resources/Exam Reviews”. Let me know if you have any questions or corrections.

Regards,

Prof. Reitz

"...how it differs from the rocks"

Hi everyone,

The exam #3 review sheet is posted under “Classroom Resources/Exam Reviews”. Let me know if you have any questions or corrections.

Regards,

Prof. Reitz

- Solve the differential equation $y' = \frac{\cos(x+y)}{e^y+\cos(x+y)}$I am trying to solve this differential equation $$\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{\cos(x+y)}{e^y+\cos(x+y)}$$ but can't reach anywhere. I was thinking an identity for cosine?
- Proof of a problem related to the derivative. [closed]Can you help me to prove part a? enter image description here Question from the image: Given that $f$ is differentiable at $c$ and let $g$ be the function defined by $$ g(x) = \cases{ f(x) & $x\le c$ \cr f'(c)(x-c)+f(c) & $x> c$ } $$ Show that $g$ is differentiable at $c$. What is […]
- Integral equation with kernel $x-y$I am trying to solve the integral equation $$ f(x)=\int_0^x (x-y)f(y)dy, \ 0\leq x\leq 1 $$ in the space $C([0,1])$ of continuous functions on $[0,1]$. My reasoning is this: Since $u(y)=(x-y)f(y)$ is continuous, the integral $\int_0^x (x-y)f(y)dy$ is a differentiable function, which means that the solution $f$ will also be differentiable. Differentiating gives $f'(x)=\int_0^xf(y)dy$. By […]
- Find a differential equation for this problemA cylinder of mass $m$ and radius $r$ is sinking in another cylinder filled with water, with radius $R$, where both of their axis are vertically positioned. Let $x$ be the height of the water respect with the level it would occupy in the absence of the small cylinder. Write the differential equation for $x$.Consider […]
- The limit is positive is the sequence bounded uniformly?Recently, I read this paper about $L^p$ Minkowski problem. In page $84$, it said that From $u_0>0$, one can see that there exist constants $m$ and $M$, s.t. $$0
- Green's function of an ODE (first order)What is the Green's Function of the following ODE (in general form)? $$ \frac{dy}{dt} = f(t) y(t) + g(t) $$
- How to find the degrees of the polynomials in this particular exponential solution?i have this differential equation: \begin{equation} \frac{dy}{dx}+e^{-2x}y^2-\frac{1}{x}(1+4x+2x^2)y = -\frac{e^{2x}}{x}(1+x+2x^2+x^3) \end{equation} and, since it is in the Riccati form, I have the following particular solution for this equation $$ y_1(x) = q(x)e^{p(x)} $$ but my problem comes when replacing in the equation. Because, it can be a bit clumsy, but I can't see how to get […]
- Linear System of Differential Equations - Why can the coefficients be variable?Consider $n$ variables $x_1, \cdots, x_n$ in a linear system of differential equations. Suppose $\forall ~i : x_i' = f_i(t,x_1, \cdots, x_n)$ Now, if all the $f_i$'s are linear functionals, then we say that we have a linear system of differential equations. If $f_i$ is a linear functional, then that means : $$ f_i(t,x_1, \cdots, […]
- Integral curve on complete surfacesI proved the following result "Prove that if $V$ is a differentiable vector field on a compact surface $S$ and $\alpha(t)$ is the maximal trajectory of $V$ with $\alpha(0)=p\in S$, then $\alpha(t)$ is defined for all $t\in\mathbb{R}.$" For every $q\in S$, there exist a neighborhood $B$ of $q$ and an interval $(-\varepsilon,\varepsilon)$ such that the […]
- ODE solution is monotonic with respect to initial value?Consider the ODE $$\frac{dy}{dt}=f(y,t), \quad y(0)=a,$$ where we may assume that $f$ is Lipschitz continuous and there is a unique solution $y(t)$ on $[0,2]$. My questions is following: Is $y(1)$ a monotone (more precisely, increasing) function of $a$ ? I believe the claim holds true because the direction fields never cross, and thus if a […]

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