The review sheet for the final exam is posted under Classroom Resources / Exam Reviews.

Regards,

Prof. Reitz

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"...how it differs from the rocks"

The review sheet for the final exam is posted under Classroom Resources / Exam Reviews.

Regards,

Prof. Reitz

- Will a forward Euler condition $ \|u + \tau f(u)\| \leq \|u\|, \quad \forall 0<\tau \leq \tau_{FE}$ indicate $\|u(t)\|\leq \|u^0\|$?Consider an initial value problem for a system of ordinary differential equations (ODEs) of type \begin{equation} \label{eqn:odesys}(I) \left\{ \begin{aligned} u_t & = f(u(t)), \quad \forall t \in (0, T], \\ u(0) & = u^0, \end{aligned} \right. \end{equation} where $u^0 \in \mathbb{R}^N$, $f: \mathbb{R} \times \mathbb{R}^N \rightarrow \mathbb{R}^N$ is a continuous function. We assume that the […]
- Which numerical method should I use that preserves solution's length?I am trying to solve an IVP problem consisting of the following equations: $y_1'=Ay_2$ $y_2'=Ay_1$ $y_3'=0$ The analytical solution considering $y(0)=\left[\begin{matrix}Csin(\theta)\\ 0\\ Ccos(\theta) \end{matrix}\right]$ is : $y_1=Csin(\theta)cos(At)$ $y_2=-Csin(\theta)sin(At)$ $y_3=Ccos(\theta)$ Clearly the length(norm) of the analytical solution is constant for different $t$. However, when I'm trying to approximate the solution using BDF or RK45 in Python […]
- Is this Cauchy-Euler DIfferential Equation solvable using reduction of order, or by the general way to solve an Euler DE?So I'm given this equation $$x^{2}y'' + xy' - y = 0$$ with the initial conditions $y(0)=0$ and $y'(0)=1$. By using the method normally used for Euler DE's, I get to the point where we have $$y(x) = c_{1}x + c_2x^{-1}$$ Except, this is where I'm getting confused, because obviously I can't divide by zero […]
- A differential equation involving the cross productIn the course of work in atomic physics, I've run into a differential equation of the form $$\frac{d}{dt}\vec{X}(t) = \vec{\Omega}(t) \times \vec{X}(t)$$ where $\vec{\Omega}(t)$ is given and we wish to solve for $\vec{X}(t)$. Is there a name for this type of equation? Can you please provide references on the structure of solutions to this equation?
- How would I solve this ODE to get in it in the form $y(x)$?$$y' = 2x + y - 5$$ I have tried doing it by separating it into $y' - y = 2x -5$ but I didn't get anywhere. I think you can do it that way(?) but you need to know about integration factors, which I don't and am not supposed to use. I tried defining […]
- Finding the general solution to a differential equation. Using C as the unknown constantUsing the formula $Y$, $y=(\frac{1}{\text{IF}})$Integrate(IF(right side) I had to solve this problem, integrate by parts, $e^{-4x}x$
- Deriving ODE as limit of recurrenceI have the following recurrence relation $$f^\Delta(t+\Delta) = \max(\alpha, e^{\beta \Delta} f^\Delta (t))$$ I would like to show that as $\Delta \to 0$, this becomes the ODE $$\max(\alpha - f(t), \beta f(t) - f'(t)) = 0$$ Here's my attempt: $$0 = \max(\alpha-f^\Delta(t+\Delta), e^{\beta \Delta} f^\Delta (t)-f^\Delta(t+\Delta))$$ If and only if \begin{align} \alpha-f^\Delta(t+\Delta) &\le 0 \\ […]
- How to reduce this equation $y=2px+x^3p^2$ to Clairaut's equation, given that $p=dy/dx$I have not got any idea about how to reduce it to Clairaut's equation . I just want to know what should i substitute in order to reduce it to Clairaut's equation of y=px+f(p). I don't need solution , i just want to know what should i substitute.
- Can I add a constant with the constant of integration?Suppose the the solution to the ordinary differential equation is $$ y = x^2 + \ln(x) + 3 + C.$$ Can I simplify it to just $$ y = x^2 + \ln(x) + C?$$
- Does Linear differential operator having factors commute?Suppose $L$ is a linear differential operator that factors but has variable coefficients. Do the factors of $L$ commute? Could someone please guide me on how to proceed in this problem?

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