# What is Euler's method: Definition and 83 Discussions

In mathematics and computational science, the Euler method (also called forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who treated it in his book Institutionum calculi integralis (published 1768–1870).The Euler method is a first-order method, which means that the local error (error per step) is proportional to the square of the step size, and the global error (error at a given time) is proportional to the step size.
The Euler method often serves as the basis to construct more complex methods, e.g., predictor–corrector method.

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1. ### I On the approximate solution obtained through Euler's method

This is a bit of a longer post. I have tried to be as brief as possible while still being self-contained. My questions probably do not have much to do with ODEs, but this is the context in which they arose. Grateful for any help. In what follows ##|\cdot|## denotes either the absolute value of...
2. ### I Solving Schrödinger's equation for a hydrogen atom with Euler's method

Hi, first-time poster here I'm a student at HS-level in DK, who has decided to write my annual large scale assignment on Schrödinger's equation. My teacher has only given us a brief introduction to the equation and has tasked us to solve it numerically with Euler's method for the hydrogen atom...
3. ### MHB 2.2.294 Euler's method word problem

George makes a paper boat with his brother Bill and goes out in the rain to play with it. It falls in the stream along the curb, racing towards the sewer. Let t be measured in seconds, p be the velocity of the paper boat in meters/second and g be George’s velocity, measured in meters/seconds...
4. ### MHB -b.2.7.2 Euler's method y'=3+y-y y(0) =1

$\tiny{2.7.2}$ 1000 (a) Find approximate values of the solution of the given initial value problem\\ at $t = 0.1, 0.2, 0.3$, and $0.4$ using the Euler method with $h = 0.1$.(b) Repeat part (a) with h = 0.05. Compare the results with those found in (a).(c) Repeat part (a) with h = 0.025. Compare...
5. ### The coming revolution in physics education

Classical physics is difficult because it is based on differential equations, and the differential equations of interest are usually unsolvable. The student must invest a lot of time in learning difficult math, and still can only analyze very simple systems. This difficulty arises in the first...
6. ### MHB Using Euler's Method to find the general solution to the diff. equation

Could someone please provide guidance on how to begin this problem? I've attached the preface to the assignment question.Show that the general solution of the differential equation y″(x)=−y(x) is y(x)=Acos(x)+Bsin(x) where A and B are arbitrary constants. Hint: You'll need the Taylor series...
7. ### MHB Solving SIR Model with Euler's Method

Question on SIR Model and using Eulers method for approximating a solution. Given the 3 ODEs of the SIR model dS/dt = -\betaSI dI/dt= -\betaSI - \gammaI dR/dt = \gammaI Ive been asked to produce in excel Eulers method for axproximate solutions. Given some initial values for S(0) and I(0) as...
8. ### MHB 242t.9.1.15 Use Euler's method to calculate the first three approximations

$\tiny{242t.9.1.15}$ $\textsf{Use Euler's method to calculate the first three approximations}$ $\textsf{to the given initial value problem for the specified increment size.}$ $\textsf{Calculate the exact solution. Round to 4 decimal places.}$ \begin{align*}\displaystyle...
9. ### MHB 9.1.14 Use Euler's method to calculate

$\tiny{9.1.14}$ $\textsf{Use Euler's method to calculate the first 3 approximations to the given initial value problem for the specified increment size.}\\$ $\textsf{ Calculate the exact solution.}$ $y'=y^2(5+5x), y(1)=-1, dx=0.2$ $y_1=$ $y_1=0.4$ $y_2=0.6268$ $y_3=1.2694$...
10. ### MATLAB "Linear Model" of a Pendulum via Euler's Method

In my problem the linear modal is defined as the first term in the series expansion of \sin(x) so: \sin(x) = x - \frac{x^{3}}{3!}+\dots \sin(x) = x is the linear modal. So with this, I then have to write \frac{d^{2}x}{dt^{2}} = -\sin(x) as a system of x^{\prime} and y^{\prime}, so...

18. ### MHB Order of accuracy- Euler's method

Hello! (Smile) Theorem Let $f \in C([a,b] \times \mathbb{R})$ a function that satisfies the Lipschitz condition and let $y \in C^2[a,b]$ the solution of the ODE $\left\{\begin{matrix} y'=f(t,y(t)) &, a \leq t \leq b \\ y(a)=y_0 & \end{matrix}\right.$. If $y^0, y^1, \dots, y^N$ are the...
19. ### MHB Euler's Method Error: Linear Functions

Hello! (Smile) Theorem: Let $f \in C([a,b] \times \mathbb{R})$ function that satifies the total Lipschitz condition and let $y \in C^2([a,b])$ the solution of the ODE $\left\{\begin{matrix} y=f(t,y(t)) &, a \leq t \leq b \\ y(a)=y_0 & \end{matrix}\right.$ If $y^0, y^1, \dots, y^N$ are the...
20. ### Use Euler's method with h=0.1 to find approximate values?

Homework Statement Use Euler's method with h=0.1 to find approximate values of the solution of the initial-value problem y'+3y=7e^(4x), y(0)=2 at x=0, 0.1, 0.2, 0.3, ..., 1.0. Homework Equations f(x, y)=7e^(4x)-3y x0=0, y0=2 The Attempt at a Solution y(0.1)=y1=y0+f(0, 2)(0.1)=2+(0,1)f(0...
21. ### Euler's method for a mass sliding down a frictionless curve

Homework Statement Consider a mass sliding down a frictionless curve in the shape of a quarter circle of radius 2.00 m as in the diagram. Assume it starts from rest. Use Euler’s method to approximate both the time it takes to reach the bottom of the curve and its speed at the bottom. Hint...
22. ### Implicit Euler's Method Application

Homework Statement dx/dt= -x2-2x(1+t+t2) x(1)=2 estimate x(1.2) with h=0.2 Homework Equations Implicit Euler: I was taught that we must solve for yk+1 using Newton's method: This doesn't seem like it will work because Newton's method assumes a function of only one variable. According to...
23. ### Use Euler's method with h=0.05 to find approximate values?

Homework Statement Use Euler's method with h=0.05 to find approximate values for the solution of the initial-value problem y'=2x^2+3y^2-2, y(2)=1 at x=0.1, 0.2, 0.3. Homework Equations None. The Attempt at a Solution Here's my work: y'=2x^2+3y^2-2, y(2)=1 f(x, y)=2x^2+3y^2-2, x0=2, y0=1...
24. ### Euler's Method to approximate a second order Differential Equation

Homework Statement y'' + 4y' + 4y = 0 ---- y(0) = 1, y'(0) = 5 Find the exact solution of the differential equation. Use the exact solution and Euler's Method to compute Euler's Approximation for time t = 0 to t = 5 using a step h=0.05. Plot Euler's & Exact vs. t and plot Error vs. t. Then...
25. ### Euler's method for coupled ODE's

Homework Statement Consider the following pair of coupled first order ODEs \dot{y_{1}} = y_{2} with ##y_{1}(0) = 1## \dot{y_{2}} = -y_{1} with ##y_{2}(0) = 1## Use the Euler integration method with a step-size ##h = 1## and fill out the entries in the table below \begin{bmatrix}...
26. ### MHB Estimate $y(0.5)$ using Euler's Method

use euler's method with step size 0.1 to estimate y(0.5), where y(x) is the solution of the initial-value problem y'=y(x+1), y(0)=1. round your answer to four decimal places. this is all I've done so far $y'=y(x+1)$ $y(0)=1$ $h=0.1$ $x_{0}=0$ $y_{0}=1$ $x_{1}=x_{0}+h=0+0.1=0.1$...
27. ### Why do we get oscillations in Euler's method of integration and what i

When using Euler's method of integration, applied on a stochastic differential eq. : For example - given d/dt v=−γvΔt+sqrt(ϵ⋅Δt)Γ(t) we loop over v[n+1]=v[n]−γv[n]Δt+sqrt(ϵ⋅Δt)Γn. (where −γv[n] is a force term, can be any force and Γn is some gaussian distributed random variable. ) . Then if...
28. ### Comparing Analytic and Approximate Solutions Using Euler's Method in Matlab

Homework Statement Hello, I am working on a problem involves my using the Euler Method to approximate the differential equation \displaystyle \frac{df}{dt} = af(t) - b[f(t)]^2, both when b=0 and when b is not zero; and I am to compare the analytic solution to the approximate solution when b=0...
29. ### MHB How can I apply Euler's method without the interval component?

I have a problem on which I need to apply Euler's method - EXCEPT that I don't have one of the crucial components. Question and my thoughts below: **Question:** Consider the initial value problem $\frac{dy}{dt}=\alpha t^{\alpha - 1}, y(0)=0$, where $\alpha > 0$. The true solution is...
30. ### Dv/dt = (v+1)/10v^2 Euler's method

Homework Statement dv/dt = (v+1)/10v^2 Euler method delta t = .5, t =0 and v = 1, to find v(4) Homework Equations I'm using v = v(old) + (stepsize)(dv/dt) I'm just wondering if anyone can confirm my results. My teacher said that although this seems easy over half the homework will...
31. ### Simple differential equation problem on euler's method? HELP

how do i find the yk value?? the k, tk and mk values makes sense to me Problem #1: http://imageshack.us/photo/my-images/14/fe0x.jpg/ Solution manual: http://imageshack.us/photo/my-images/5/hu55.jpg/
32. ### Question on Euler's method - Calculations using rounded numbers? HELP

Homework Statement Taking step size h = 0.2, use Euler’s Method to determine y(1.6), given that dy/dx = ln(2y+x) ; y(1)=1.2 Record your results to 5 decimal places at each step. Homework Equations N/A The Attempt at a Solution My question is to do with the method, not the...
33. ### Solving Euler's Method: Accuracy of y_100 = 3.1515

Homework Statement http://img853.imageshack.us/img853/5578/euler.png Homework Equations y_(n+1)=y_n+f(x_n,y_n)h The Attempt at a Solution With n = 100 and step size h = 0.01, I got y_100 = 3.1515 I don't see how this is right, because the directions imply that it's supposed to be accurate...
34. ### Differential Equations Euler's method

Find the solution y = φ(t) of the given problem and evaluate φ(t) at t = 0.1, 0.2, 0.3, and 0.4. 1.y'=3+t-y y = φ(t)=t-2e^-t y(1)= 0+(0-2e^0)*(.1)=.8 and the correct answer is 1.19516 2. y'=2y-1 What I'm getting stuck on is do I use the formula y(n)=y(n-1)+f(t(n-1),y(n-1)h because...
35. ### DE Derivative Using Euler's Method

Homework Statement The velocity v of a skydiver is well modeled by a differential equation: m*dv/dt = mg - k*v^2 Where m is the mass of the skydiver, g = 9.8 m/s^2 is the gravitational constant, a k is the drag coefficent determined by the position of the diver during the dive. Consider a...
36. ### Euler's Method of Trigonometric Series- Real Analysis

Homework Statement Use Equation (1) to derive the formula: (1-a*cos(phi))/(1-2*a*cos(phi)+a^2))= 1+a*cos(phi)+a^2(cos(2phi))+a^3(cos(3phi))... Homework Equations Equation (1) is 1/(1-a(cos(phi)+isin(phi)))= 1+ [a(cos(phi)+isin(phi)]+ [a(cos(phi)+isin(phi)]^2 +...
37. ### What is the approximate value of f(0.98) using Euler's method?

Homework Statement consider the function f(x) = aln(x+2). Given that f'(1) = a/3, what is the approximate value of f(0.98)?Homework Equations f(x1) = f(x0) + f'(x0)x(x1-x0)The Attempt at a Solution I solved it and get f(.98) = aln(1+2) + (.098-1) = aln(3) - (.02)(a/3) <= not an answer the...
38. ### MHB Is this the Correct Equation for Approximating y(0.8) Using Euler's Method?

OK, my homework system says that the 0.20000 is correct... but all of my others are incorrect... Is this the correct term for the top right box?:confused: y(0.8)=y(0.4)+(\Delta{t} \times y^{\prime}(0.4)) OR y(0.8)=y(0.4)-(\Delta{t} \times y^{\prime}(0.4)) According to the 0.20000 answer...
39. ### Stability of an ODE and Euler's method

I have been thinking about numerical methods for ODEs, and the whole notion of stability confuses me. Take Euler's method for solving an ODE: U_n+1 = U_n + h.A.U_n where U_n = U_n( t ), A is the Jacobian and h is step size. Rearrange: U_n+1 = ( 1 + hA ).U_n This method is...
40. ### Numerical Approximations - Euler's Method

Homework Statement Here is the problem The Attempt at a Solution I was able to draw the directional field and found which regions had a positive or negative slope. However I don't get what the question means by "Observe that there is a critical value of α in the interval 0 ≤ α ≤ 1 that...
41. ### MATLAB Euler's Method in MATLAB: Solving for dy/dt = cos(y)

Hello I have a program for Eulers method >> % Euler's Method for dy/dt = cost k = 1; y0 = 0; npoints = 500; dt = 0.01; y = zeros(npoints,1); % this initializes the vector y to being all zeros t = zeros(npoints,1); y(1) = y0; % the initial condition t(1) = 0.0; for...
42. ### Exploring Euler's Method with Δt = 0.25

Homework Statement Use EulersMethod to perform Euler's method with the given step size Δt on the given initial value problem over the time interval specified : dy/dt= (y^2)-4t , y(0)=0.5 , 0<=t<=2, Δt=0.25The Attempt at a SolutionThis is what I did but I don't think its right, because in the...
43. ### Determine n for Ensuring Global Error is Less Than 10^-2 in Euler's Method

Homework Statement Use Euler's method with h = 1/2 to estimate y(1) for the IVP: y(0)=1 y'(t)=t^2-y(t) Assuming that |y(t)| \le 1 for 0 \le t \le 1 determine the value of n needed to ensure that |E_n| \le 10^{-2} Homework Equations |E_n| \le \frac{T}{L}(e^{L(t_n-t_0)-1}) The Attempt...
44. ### How Does Euler's Method Help Predict the Ball's Landing in Matlab Simulation?

Homework Statement Question : The ball is kicked at a 5m high vertical wall 20m away. The initial speed of the ball is Vm/s and its initial angle of motion is C degrees with the horizontal (both the speed and angle are inputs). When the ball strikes the wall it bounces back with...
45. ### Integrating Euler's equations for rigid body dynamics with Euler's Method

Hello, I am trying for a couple of hours now to integrate these equations ( http://en.wikipedia.org/wiki/Euler%27s_equations_%28rigid_body_dynamics%29 ) with the Euler's method: \dot{f}=\partial{f}/\partial{t}\cong\Deltaf/\Deltat=(f(t+\Deltat)-f(t))/\Deltat . I am trying to do this...
46. ### Forming a matrix using Euler's method and ODE

L is the operator. Lx=x'(t)+u(t) x(t) =0. Provided that x(t0)=x0. Before writing the matrix. The book express it out in equations. x(t0)==x0 x(t1)-x(t0)+Δt u(t0) x(t0)==0 x(t2)-x(t1)+Δt u(t1) x(t1)==0 ... Euler's method is x(t0)+Δt f[x0,t0], right? so where did the x'(t) from the...
47. ### Projectile motion with air resistance using Euler's method

Homework Statement Calculate the trajectory of our canon shell including both air drag and reduced air density at high altitudes so that you can reproduce the results in Figure 2.5. Perform your calculation for different firing angles and determine the value of the angle that gives the...
48. ### Implicit (Backwards) Euler's Method

How would one go about solving for one variable for an implicit Euler's method such as this: I am completely lost...all I know is the value of U and dT Un+1 = Un + (dU/dT)|n+1dT Vn+1 = Vn + (dV/dT)|n+1dT
49. ### Compute the velocity of a free-falling parachutist using Euler's method

Homework Statement For the second order drag model (Eq. 1.8), compute the velocity of a free-falling parachutist using Euler's method for the case where, m = 80 kg Cd = .25 kg/m Perform the calculation from t = 0 to 20 with a step size of 1 s. Use an initial condition that the parachutist...
50. ### Anyone can describe how euler's method works?

let's say we do not know the proof. then how do you guys describe it conceptually? hm...><