Assignments Week 9

March 25, 2019 / Jonas Reitz / 0 Comments

WeBWorK: Assignment 9-Nonhomogeneous due Tuesday, 4/2, at midnight.
NOTE: Because of our exam on 4/2, assignment 10-Trench-ReductionOfOrder will be due the following Tuesday, 4/9.

OpenLab: OpenLab #2: Numerical Methods PROJECT is due Thursday, 4/4, by the start of class.

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When curvature is proportional to inverse square of ordinate
After integration of (differentials primed w.r.t. $x$ ) $$ \frac{y''}{{(1+y^{'2})^\frac32}}=\frac{c_1}{y{^2}}$$ The general solutions seem to be the catenaries only $$ y={c_1\cosh \frac{x}{c_1}}+c_2 $$ Is it correct?
A very weird integral equation
I tried to come up with an integral equation for fun, and made this creature:$\def\d{\mathrm d}$ $$ f(x)-\int_x^{2x}f(t)\,\d t=0, $$ so I followed these steps: $$ \begin{aligned} &f(x)+\int_x^{2x}f(t)\,\d t=0\\ \implies\quad&\frac{\d f}{\d x}+2f(2x)-f(x)=0\\ \implies\quad&\frac{\d f}{\d x}-f(x)=-2f(2x)\\ \implies \quad&\left(\frac{\d}{\d x}-1\right)f(x)=-2f(2x)\\ \implies\quad&f(x)=\frac{2f(2x)}{1-\frac{\d}{\d x}}\\ \implies\quad&-f(x)=2f(2x)\sum_{n=0}^{\infty}\frac{\d^n}{\d x^n}=2f(2x)\left(1+\frac{\d}{\d x}+\frac{\d^2}{\d x^2}+\ldots\right)=2f(2x)\\ \implies\quad&f(x)=2f(2x). \end{aligned} $$ But since I am but a humble mathematical […]
Error in deriving Rodrigues Formula for Legendre Polynomials
I'm trying to derive Rodrigues Formula for Legendre Polynomials "from scratch". I see related questions typically starting where I want to end up. I think I'm going wrong on the last step, but I'm not sure how. I'd like to derive the RF without Generating Functions, just orthogonality (up to multiplicative constants) leading to a […]
Is integration based on parametrization of the function something commonly used? Is there merit in doing so without ever partitioning the x-axis? [closed]
A Geometric Approach to Numerical Integration A definite integral can be approximated by walking the integrand at uniform speed and projecting onto (or reading off of) the x-axis. For each step, we can use subtraction to tell if we've got higher, lower or stayed level. This displacement information can be used with our stride length […]
The solution of ODE using Frobenius' Method, how to explain its convergence?
How to state the convergence of the solution of this ODE and what is the solution? Also, Is there online site to check its solution so that I could analyse it whether my answer is correct, MatLab, Wolfram, anything? ODE: $$9x(1-x)y''-12y'+4y=0$$ writing it in this form: $$x^2y''+xp(x)y'+q(x)y=0$$, I got $p(x)$ and $q(x)$, $\frac{12x}{9(1-x)}$ and $\frac{4x}{9(1-x)}$ […]
Proof of Euler identity using ODE [closed]
I am writing an article to prove Euler identity :$e^{i\pi}+1=0$ Here the main part: Consider the function :$ \mathbb{R} \rightarrow \mathbb{C}, f(x)=e^{ix}$ Differentiating twice,we get : $f''(x)=-f(x)$ So the function $f$ satisfies the linear 2nd order ODE : $y''+y=0$,which has general solution $y= a\cos x + b\sin x$,where $a$ and $b$ are complex constants Is […]
Lyapunov Approximation of Separation of Trajectories
I am self-studying dynamical systems, and I came across a property that I am unsure if I correctly identified how it is found. That is, $$ ||\phi_t(x+\epsilon) - \phi_t(x)|| \approx ||\epsilon|| e^{t\lambda} $$ Correcting the original property. I misread the statement. But it looks like they're stating the difference is approximately the length of the […]
Solving second order differential equation graphically
I am stuck on the following problem, hoping that someone will be able to help me. I have a following second order differential equation: $$y''=-\sqrt{y}+0.5y'$$ with the following initial conditions: $$y(0)=0, y'(0)=0$$ I am familiar with Euler's method for first order differential equation when $y'=f(x,y)$, but In the above case I get stuck. When I […]
Is it plausible to express $1\cdot 5\cdot 9\cdots(4n-3)$ and $7\cdot 11\cdot 15\cdots(4n+3)$ into a single factorial?
This came up when I tried to solve an ODE: $4xy''+y'-y=0$ using Frobenius method. I'm stuck at rewriting this into a compact form: $$z(x,0)=a_0\left[1+\frac{x}{1\cdot1}+\frac{x^2}{1\cdot2\cdot1\cdot5}+\cdots+\frac{x^n}{1\cdot2\cdot3\cdots n\cdot1\cdot5\cdot9\cdots(4n-3)\cdots}\right]$$ $$z\left(x,\frac34\right)=a_0\left[1+\frac{x}{\frac74\cdot4}+\frac{x^2}{\frac74\cdot\frac{11}{4}\cdot4\cdot8}+\cdots+\frac{x^n}{\frac74\cdot\frac{11}{4}\cdots\frac{4n+3}{4}\cdot 4\cdot8\cdot12\cdots4n\cdots}\right]$$ What I want is something like this: $$1\cdot3\cdot5\cdots(2n-1)=\frac{(2n)!}{2^n n!}$$ But no matter how, I couldn't manage to get a beautiful compact form using only a single factorial (If […]
Differential equation $yy^\prime-7y=6x$ [closed]
I have the differential equation $yy' - 7y = 6x$, and I know that it's a homogeneous equation. However, when I try to solve it, I end up with $\int \frac{-v}{v^2 - 7v - 6} dv$, which is a difficult integral. Can you help me find an integrating factor or another solution for this?

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Assignments Week 9

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