Tagged Questions

4
votes
2answers
53 views

Does this ODE have an exact or well-established approximate analytical solution?

The equation looks like this: $$\frac{\mathrm{d}y}{\mathrm{d}t} = A + B\sin\omega t - C y^n,$$ where $A$, $B$, $C$ are positive constants, and $n\ge1$ is an integer. Actually I am mainly concerned ...
1
vote
0answers
34 views

Small Inhomogeneity of Differential Equation

Given a variable $x\in[-L,L]$ with $L\in \mathbb{R}$, first consider a generic homogeneous second order differential equation with potential $V(x)$: $$\left(\frac{d^2}{dx^2}+V(x)\right)f(x)=0$$ Let ...
1
vote
2answers
32 views

Formula for the pseudofrequency using approximations

We know that $w_1=\frac{\sqrt{4w_{0}^2-b^2}}{2}$ and $(1+u)^{\alpha}=1+\alpha u + \mathrm{O}(u^2)$ I need to show that $\frac{w_1}{w_0}=1+\frac{\lambda}{N^2}+\mathrm{O}(\frac{1}{N^3})$ such that ...
2
votes
0answers
30 views

Approximate Differential Equation?

Let $x \in \mathbb{R}$ be a variable and $c\in\mathbb{R}$ a parameter. Also, let $f(x,c)$ be a function dependent on $x$ and $c$. Furthermore, define a Differential Equation which is solved by the ...
1
vote
0answers
44 views

Intuition for approximating Ei(x)

I'm working with a function that is defined to be the sum of a series, and I'd like to either find its values, or approximate them analytically: $F(x) = \frac{1}{w} ...
1
vote
1answer
171 views

Method of dominant balance

Find the leading asymptotic behaviour as $x \rightarrow \infty$ of $$x^2y'' + (1 + 3x)y' + y = 0 $$ Can someone kindly explain me how to solve this problem? Im learning asymptotic analysis, and I ...
2
votes
2answers
82 views

Using differentials to approximate a function

So I have a homework problem that I cannot figure out. I am supposed to approximate the value of $\sqrt{(4.98)^2-(3.03)^2}$ using differentials. What I have so far is $$f(x,y)=\sqrt{x^2-y^2}$$ ...
0
votes
0answers
41 views

Finite difference approximation for two terms

How can I discretize the following using finite difference approximations? $$\frac{\partial}{\partial x}\left[\left(\frac{\phi D_w}{\tau}+\frac{\alpha k(x,y,t)}{\mu}\left(\nabla p(x,y,t)-\rho ...
0
votes
1answer
107 views

Soundwaves under the water

I've got the following problem that is taken from the numerical analysis book by Kahaner-Moler-Nash (P8-15): The speed of sound in ocean water depends on pressure, temperature and salinity, all ...
0
votes
0answers
47 views

A question applying Ritz Galerkin.

Let S = (0,1)^2 contained in R^2. If we see the boundary value problem -(Triangle)u = (pi)^2 cos(pi*x1) in S, del(subscript n) u = 0 on the boundary, we have to 1) Provide a functional J, where ...
1
vote
1answer
291 views

Picard's method application

Hello guys here is my question thank you for all the help. I need to determine On which integral the Picard's Method is applicable for $y'=xy^2$, $y(0)=0$ and need to calculate the first ...
3
votes
0answers
159 views

Approximating a system of differential equations as a Bézier curve

I am looking for a general transform to approximate the solution to an n-dimensional system of differential equations and initial conditions as a cubic or quadratic Bézier curve. Sorry if my ...
1
vote
1answer
49 views

Euler's approximation of $m' = -\frac{m}{V}v$

Continuing my previous question from today - I should code a program that finds approximate value of the aforementioned differential equation. The full text of the assignment is: Water purifier with ...
2
votes
1answer
187 views

Euler's method approximation

I'm supposed to write a program for approximating the value of function $y = y(x)$, which is given as: $$y' = \frac{1+y}{1 + x^2}$$ I also know that $y(0) = 0$. I should approximate the value for ...
3
votes
1answer
465 views

With Euler's method for differential equations, is it possible to take the limit as $h \to 0$ and get an exact approximation?

I was recently watching a tutorial on Euler's method for approximating differential equations, and the whole time I was thinking "why can't you just take the limit of the step size $h$ as it goes to ...
3
votes
1answer
380 views

Large Deviation Properties of a function of a geometric random variable

Suppose I have a variable $s$ that has a geometric distribution with success parameter x. So the probability of success on trial $s$ is $p_s = (1 - x)^{s - 1} x$, Consider the following function of ...