Assignments Week 3

February 14, 2019 / Jonas Reitz / 0 Comments

WeBWorK: Assignment 5-Exact Due Tuesday 2/19 at midnight
NOTE: As announced in class today, the WeBWorK assignments from last week have been extended until next Tuesday, 2/19 at midnight:
3-Separable , 4-Bernoulli-nosub and 4b-Homogeneous-YoverX due Tuesday 2/19 at midnight.

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assignments week 3

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Finding the Steady-State Solution of the Burgers' Equation
I am trying to determine the steady-state solution of the following Burgers' equation: \begin{equation*} u_t+u u_x=\nu u_{x x}, \quad u(x, 0)=\sin (\pi x) \end{equation*} subject to the boundary conditions $u(0, t)=u(1, t)=0$. Could someone provide guidance on how to find the steady-state solution for this equation? Any tips or references for solving such problems would […]
Second order differential equation with ln
I have a differential equation I need help solving $$f''(x)-(a^2+B\ln(x))f(x)=0$$ where a and B are constants. I know f(x) must be e to the power of something but no exponents works correctly. I got the equation from a Schrödinger equation after I plugged in the potential energy. Is there even an analytical solution? I'm not […]
Solutions of an ode with Wronskian zero?
It is well known that in order to check linear independence (dependence) in a set of solutions of an ode, one computes the Wronskian of such functions and, if it is different from zero, the solutions are linearly independent (li). It is also known that there can be li functions whose Wronskian is zero, even […]
Solving the coupled Nonlinear System in Stability analysis using Lyapunov function [closed]
I'd like to ask about the stability of system using Lyapunov function. How to find an appropriate Lyapunov function to show the stability of a system: $$ \begin{align*} \dot{S}(t) &= k - \frac{\mu_{\text{max}} S(t) b(t)}{\delta \left(K_m + S(t)\right)} + \frac{d \beta b(t)}{\delta} + \frac{\alpha \beta b(t)}{\delta}, \\ \dot{b}(t) &= \frac{\mu_{\text{max}} S(t) b(t)}{K_m + S(t)} - […]
Which of the solution curve of the differential equation is correct? [closed]
curve 1 enter image description here curve 2 enter image description here Which one is correct? Why? Any tips to draw such solution curve?
How to integrate this differential equation?? [closed]
I encountered a differential equation while proving energy conservation in physics but was unable to integrate it. $\frac{dv}{dt}=pv-qv^2t$ Here $p,q \in R$ I tried converting it to linear form of differential equation but was unable to do so. Any help is appreciated!!
Strange ODE system
As I was solving a physics problem I came across this very strange ODE. My goal is to get the differential equation for the temperature T. The problem is that I can't get rid of the current I in any ways. Thanks $$\begin{cases} I=\dfrac{\alpha}{R_0}(T-T_0)-\dfrac{L}{R_0}\dfrac{dI}{dt}\\ C\dfrac{dT}{dt}=-\alpha TI-k(T-T_0) \end{cases}$$ I tried differentiating the equations with respect to […]
Can we interpret indefinite integrals as limit of sums like definite integrals? If not, how does solving separable differential equations make sense?
To illustrate my point, let's consider an example: the amount of a radioactive substance is a function of time, $Q(t)$. At the beginning, $t = 0$, and the initial amount of the radioactive substance is $Q(0) = Q_0$. The substance decays at a rate \begin{equation} \frac{dQ}{dt} = -kQ \end{equation}, where $k$ is a positive constant […]
Best Textbook for Applied Ordinary Differential Equations
I'm looking for suggestions for a textbook that covers Ordinary Differential Equations and their applications, I understand this is a commonly asked question, but I'm looking for a text that would best supplement the course I'm taking, which covers: First Ordinary Linear Differential Equations Dimensional Analysis and Buckingham Pi Theorem Second Order Linear Differential Equations […]
Application of Picard-Lindelof Theorem to enumerate all solutions of y'=y^2.
How to use Picard-Lindelof Theorem to show that all solutions of $y'=y^2$ on some open interval $I$ bounded in one-side are either of the form $y=\frac{1}{-x+C}$ (where $C$ is any constant that is not in $I$) or $y=0$? And if the interval is the whole real line then the only solution is $y=0$? What I […]

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Assignments Week 3

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