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.

Assignments, Uncategorized

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Generilezed Clairaut Equation
I read about Clairaut's Generalized ODEs (from an exercise sheet) which have the following form: $$f(xy'-y)=g(y')$$ I have shown that: $y''[xf'(xy'-y)-g'(y')]=0$ From there i can easily show that the following family of function: $$y(x)=Ax+B$$ satisfy this ODE. But i do not know how to find a solution for the equation: $$xf'(xy'-y)-g'(y')$$ Is there a general […]
How to calculate the derivative of a function of an ODE flow?
Let's consider a differential equation in $\mathbb{R}^n$: $$ \dot{\mathbf{x}} = f(\mathbf{x}) $$ with $f$ smooth or even analytic. Let also $\varphi^t(\mathbf{x})$ be the flow of the differential equation. Given a non-negative function $h:\mathbb{R}^n\longrightarrow\mathbb{R}^+$, let's define another function $g:\mathbb{R}^n\times\mathbb{R}^+\longrightarrow\mathbb{R}^+$ $$ g(\mathbf{x},t):=h(\mathbf{x}-\varphi^t(\mathbf{x})) $$ I now am trying to calculate the derivatives of $g$, in particular $$ \frac{dg}{dt},\qquad\frac{\partial […]
Implementing Initial Conditions in Autonomous Differential Equation
I have the differential equation $ yy'' - 5(y')^2 + y^2 = 0$ with initial conditions $y(0) = 1$ and $\frac{dy}{dx}(0) = 0$ and have been asked to use reduction of order to solve it. With the substitution $v = y'$ and $v' = v \frac{dv}{dy}$, we get the Bernoulli differential equation: $$\frac{dv}{dy} = \frac{5v}{y} […]
Converting a non-markovian system to a Markovian system [closed]
A system being Markovian is more a property of the model used rather than the underlying phenomena. As an example, in classical mechanics The system described by the position x(t) is generally non-Markovian, as the equations describe movement also depend on the derivative x’(t) The system described by the tuple (x, x’) is Markovian Any […]
Construction of general solutions from a particular solution. Ermakov equation.
My question is in particular about the Ermakov equation, but in general it is about the construction of general solutions to ordinary differential equations. This is the first time I see such construction, so apologies if it is too trivial. The Ermakov equation $$ \frac{d^2y}{dx^2} + f(x) y = a y^{-3}, $$ where f(x) is […]
Global asymptotic stability for difference equations
I tried to ask this question to the economics forum but no one was able to answer me. It regards the global asymptotic stability of the Solow model. But, I'll make the question as plain as possible without any reference to economics. Consider the following difference equation: \begin{equation} k_{t+1} \equiv g(k_t) = s f(k_t) + […]
How to tackle the Laplace Transform of the $n$th derivative if we don't know $f^{(k)}(0)$ from boundary conditions?
The Laplace Transform for $n$th order derivatives is $$\mathcal L\{y^{(n)}(t)\}=s^nF(s)-\sum_{k=0}^{n-1}s^ky^{(n-1-k)}(0),$$ wherein there appear $\{y(0),y'(0), y''(0), ..., y^{(n-1)}(0)\}$. My question is basically how can we cope with boundary conditions that don't include some or any of these terms? Maybe we could redefine the independent variable in some way? Or maybe we can't do anything about it?
First order non-homogeneous ODE - variation of parameters confusion
How did they get the circled part of the textbook (page 250 ODEs by William F. Trench)? Here are my notes on the variation of parameters method: ^^ following this method for the example above, should it be $z=vx \implies z'=v'x+vx'$, So, substituting $z,z'$ back into the original ODE, we get: $$z'-\frac zx = v'x+vx'-\frac{vx}{x}=v'x+vx'-v=xe^{-x},$$ […]
Here is a title question for the revised question: "Algebraic Representation of Differential Forms: Unpacking the Notation and Integration" [closed]
"Represent a 2-form $\omega = f dx \wedge dy$ as a sum of terms, each consisting of a scalar function multiplied by a wedge product of basis 1-forms. Then, explain how this representation relates to the integration of $\omega$ over a region $\Omega$, and how the volume form $dV$ encodes the orientation and volume of […]
Convert this DE into homogenous [closed]
I was given these two differential equations under a Homogenous DE lecture and was asked to solve it. How do in convert these DEs into homogenous DEs? $$ (2x-y-3)dx-(x+4y+3)dy=0 $$ $$ (2x-3y+2)dx+3(4x-6y-1)dy=0 $$

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Assignments Week 9

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