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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

- I can't solve this second order non linear ODE, please helpCan some solve the following equation or at least tell me how to approximate the result? $$y''=a\frac{y}{(y')^2+b},y(0)=0,y'(0)=c,\forall c\in\mathbb{R}$$ I have no idea if it is an easy task, but for my level of math, I can't solve it and I need it for a project. Wolfram cannot give me an answer, at least without pro […]
- Bijectivity and smoothness of nonautonomous ODE solutionLet $f: \mathbb{R}^n \times [0, T] \rightarrow \mathbb{R}^n$, and define the ODE: $d x(0) = f(x(t), t) dt$ with initial condition $x(0) = x_0$. Assuming a unique solution, the solution characterizes a map $x_0 \mapsto x(T)$. I am looking for conditions on $f$ which make the map $x_0 \mapsto x(T)$ a diffeomorphism, or at least […]
- Checking the Existence of solution to the DE given initial valuesThe ODE is as given below: $$\frac{dy}{dx}=\frac{yx^2-xy^2}{x^3};y(0)=0$$ Following the rule, we take $f(x)=\frac{yx^2-xy^2}{x^3}$ Checking the continuity of $f(x)$, we have; $$\lim_{(x,y)\to(0,0)}{\frac{yx^2-xy^2}{x^3}}$$ Converting x and y to polar co-ordinates, $$\lim_{r\to0}{\frac{\sin(\theta)(\sin(\theta)-\cos(\theta))}{\cos^2(\theta)}}$$ This limit does not exist as it varies for different value of $\theta$. This means that the solution should not exist for the given initial values. […]
- Prove that the solution to a cauchy problem is monotonically decreasingConsidering the cauchy problem \begin{cases} x' = 5\cos^2 (tx)-x^2 -5\\x(t_0)=x_0 \end{cases} How can i prove that if $x_0 \neq 0$ then the solution $x(t)$ is monotonically decreasing and then find the solution when $x(t_0) = 0, t_0 \in \mathbb{R}$
- How to find a particular solution of a differential equation with a fractional function of x?I am finding $y$ that satisfies $$y^{(4)}-81y = \frac{12}{x^5}-\frac{40.5}{x}.$$ First, we can easily find $y_h$ as follows: The characteristic function can be derived as $\lambda^4 - 81$=0. From the characteristic function, we have $\lambda = \pm3$ or $\lambda = \pm i3$. Hence, we have $$y_h = c_1 e^{3x} + c_2 e^{-3x} + c_3 \cos(3x) + […]
- Find the definite solutionFind the definite solution of the following. Given $$ \begin{cases} C_t= 300+0.5Y_t + 0.4Y_{t-1}\\ Y_t= 200+0.2Y_{t-1} \\ Y_o= 65~000 \end{cases} $$
- Describe bifurcation for $\mu \in \mathbb{R}$ and draw the diagramI am working on the question that describe bifurcation for $\mu \in \mathbb{R}$ and draw the diagram. $$ % \begin{align} % \dot{x} &= -x^4 +5\mu x^2+4\mu^2:=f(x,\mu) \\ % \dot{y} &= -y \\ % \end{align} \tag{1} % $$ Note that $f(x,\mu)=0$ is that $x^2=\frac{5}{2}\mu\pm\frac{\sqrt{41}}{2}|\mu|$ I first obtain equilibrium points: (1)$\mu>0$, equilibrium points are $x_{1,2}=\mp\sqrt{\frac{5+\sqrt{41}}{2}\mu}$ Note that […]
- Legendre Differential equation, n(n-1) or n(n+1)I am confused regarding the Legendre Differential Equations' coefficients. In some books its, $(1-x^2)y''-2xy'+n(n-1)y=0$ and somewhere it is, $(1-x^2)y''-2xy'+n(n+1)y=0$ what is its correct form?
- How to show that is unique asymptotic stableBased on this question:Poincaré-Bendixon show periodic solutions. Show that the system $x^{'}=x-y-x^{3}$,$y^{'}=x+y-y^{3}$ has a unique periodic orbit on annulus $A:=\{(x,y): 1\le x^2+y^2\le 2\}$ and this periodic solution is asymptotic stable. I consider function $V(x,y)=(x^2+y^2)/2$ and show that $\dot V(x,y)>0$ on the circle $x^2+y^2=1$ and $\dot V\le 0$ on $x^2+y^2=2$. So $A$ is positively invariant. By […]
- Find a function that does not satisfy that IVPFind $f\in C^{\infty}(\Bbb R,\Bbb R)$ s.t IVP has not satisfied $f(x')=0$ and $x(0)=0$. Try: I found a paper that talks about that and gives an example but from $\Bbb R^2$ to $\Bbb R$, on page 347, I would like how to take that function and pass it from R to R or something like that […]

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