Assignments Week 2

February 5, 2019 / Jonas Reitz / 0 Comments

NOTE: City Tech will be closed next Tuesday, 2/12/19, for President’s Day.

WeBWorK: 3-Separable , 4-Bernoulli-nosub and 4b-Homogeneous-YoverX. Due Thursday 2/14 at midnight
OpenLab: Complete your first OpenLab Assignment “Advice from the Past” by Thursday, 2/14, at the beginning of class.

Assignments, Uncategorized

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Equivalents Phase Space
Let $f :\mathbb{R}\to \mathbb{R}^3$ be a smooth function, and $x = (x_1, x_2, x_3)$ a solution of the autonomus system $$ x' = f(x).$$ Suppose the phase space of this system is $\Theta = \mathbb{R}^3$ and assume that I already some informations about it. Now, I would like to add some hypoteses about $x$. For […]
Weak formulation for $-u''+u=1$
I am trying to solve the following problem. Now, based on other examples I have seen, it seems the weak formulation should be something in terms of an inner product. In particular, we see $$(-u+1,v)=(-u'',v)=\int_0^1(-u''(x))v(x)dx=(-u'(x)v(x))\vert_0^1+\int_0^1 u'(x)v'(x)dx=(u',v')$$ assuming $v(0)=0$ and $u$ is a solution to the ODE. So I believe the weak formulation should be to […]
How to use Green's functions to solve $y^{\prime \prime}-\pi^2y=\cos(\pi x)$, $y(0)=y(1)$, $y'(0)=y'(1)$
I have been trying this problem for several days. It is impossible for me to find the Green function of this problem!!! I don't know if it's because it's done differently or because I'm making silly mistakes. In my tiredness and desperation, I am asking you. Thank you very much in advance. Other exercises of […]
$2$nd Order Nonlinear ODE with $y'^2$ Term
I am attempting to find a closed-form solution to the differential equation $5y'^2 - 4yy'' = 0$. The substitution $y' = v$ does not appear to work here, due to the $y'^2$ term on the left hand side. How would I go about approaching this problem?
Closed Formula for maximal interval of existence of autonomous ODE
I just stumbled across the following theorem (https://jordanbell.info/LaTeX/mathematics/domainODE/domainODE.pdf) and was wondering about its validity and proof, as the author does not provide a reference. And it does seem like a very strong and useful result. Assume we have an autonomous ODE: \begin{equation*} \frac{dx}{dt} = f(x),\: x(0) = x_0 \end{equation*} for a continuous, positive function $f:[x_0, […]
Particular solution $x'=Ax+\cos(\alpha t)d+\sin(\alpha t)d $
Let be $A\in R^{n \times n},x_0,c,d\in R^n$ and $\alpha \in R-\{0\}$. Prove that the system $$x'=Ax+\cos(\alpha t)c+\sin(\alpha t)d $$ $$x(0)=x_0$$ has a particular solution of the form $\varphi_p(t)=\cos(\alpha t)v+\sin(\alpha t)w$ where $v,w\in R^n$ are constants, if and only if $$(A^2+\alpha^2 I)x_0= -(Ac+\alpha d)$$ Is this problem correct? My attemp: From the system we can obtain […]
Nowhere analytic smooth solutions to analytic differential equations
There are several parts to this question. Clearly there exist "analytic" differential equations of a certain variety that have nowhere analytic solutions (e.g., $f'(x)=2f(2x)$ has the Fabius function as a solution and $f'$ is analytic in $f(2x)$). I want to know in more restricted cases whether differential equations that have convergent Taylor series expansions in […]
Differential Equation $\frac{dy}{dt} = \frac{y}{y+t}$
I’m trying to solve the differential equation $\frac{dy}{dt} = \frac{y}{y+t}$, but I am not sure how to approach this. I have tried to substitute $u = y+t$, but this did not get me anywhere. Thank you!
Closed form for a differential equation with sin
I am currently trying to model some dynamics for a project of mine. But I got following differential equation: $$\frac{d^2\theta}{dt^2}=a\theta+b\cdot \sin(\theta-c)$$ Where $a,b,c,\theta \in \mathbb{R} $ I know how to solve this numerically and tecnically that is also enough, but it would be really neet to have a closed form solution for this problem. Does […]
Fitting a parabola through two points, minimizing the error in the derivative in those two points.
Let $f(x)$ be a parabola and $f'(x)$ its derivative. Given are the known points $(p, q)$ and $(r, s)$: $q = f(p)$ and $s = f(r)$. Also given are the values of the desired derivatives in $p$ and $r$. Lets say $t$ and $u$. In the ideal case, we want to find an $f$ such […]

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Assignments Week 2

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