1
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0answers
43 views

Approximating an IVP

I wish to solve the IVP: \begin{align} x(0) =& -1 \\ x' =& 1 + x^2 - t^3 \end{align} With a fourth order taylor series method, I solved the ODE on the interval [0, 2] and then made the ...
1
vote
0answers
56 views

Calculate a 5x5 Vandermonde system for a 5 point mesh

This is problem 1.2 in Randall J Leveque's textbook, "Finite Di fference Methods for Ordinary and Partial Di fferential Equations". I'm struggling with how to actually do the computation, I'm not so ...
3
votes
2answers
2k views

Explanation and Proof of the fourth order Runge-Kutta method

Runge-Kutte 4th order method is a numerical technique used to solve ordinary differential equation of the form $dy/dx=f(x,y), y(0)=y_0$ It gives $y_{i+1}$ in the form $y_{i+1} = ...
0
votes
2answers
178 views

Approximation using Euler's method.

Consider the initial value problem $$\dfrac{dy}{dx} = y,y(0) =1$$ Approximate $y(1)$ using Euler's method with a step size of $\dfrac{1}{n}$, where $n$ is an arbitrary natural number. Use this ...
1
vote
1answer
46 views

Show that the method is of order 3 if a =-5 and of order 2 if a is not equal to -5

I am studying ODEs and came across this exam question: I have the solution here also: I have been working on this exam question all day and have been stuck for hours. What I don't understand is ...
0
votes
1answer
76 views

truncation error - help

I am trying to understand the concept of local truncation error and came accross this in my lecture notes: what I don't understand here is where the term 'O' comes from and what it stands for in ...
1
vote
2answers
98 views

Approximation of differential equations

Can someone provide me a good reference about approximation techniques in continous domain (not piecewise nor numerical methods) for differential equations?
4
votes
2answers
99 views

Approximation of DE

It depends on my previous question. Closed form solution of DE I don't want to deal with Airy functions. How can I approximate this DE in continous domain $[0,1]$? $$y''(x)+(x+1)y(x)=0\quad\text{ ...
4
votes
2answers
64 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
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0answers
42 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 ...
2
votes
2answers
42 views

Formula for the pseudofrequency using approximations

We know that $w_1=\frac{\sqrt{4w_{0}^2-b^2}}{2}$ and $(1+u)^{\alpha}=1+\alpha u + \mathrm{O}\left(u^2\right)$ I need to show that ...
2
votes
0answers
43 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
67 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} ...
2
votes
1answer
503 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
225 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
1answer
178 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 ...
1
vote
1answer
437 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
200 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
50 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
254 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
672 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 ...
4
votes
1answer
441 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 ...