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Eulers Method Calculator

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Example
Created on 2024-06-20Asked by Abigail Allen (Solvelet student)
Use Euler's method with a step size of 0.1 to approximate the solution of the differential equation dydx=x+y \frac{dy}{dx} = x + y with initial condition y(0)=1 y(0) = 1 at x=0.5 x = 0.5 .

Solution

To approximate the solution of the differential equation dydx=x+y \frac{dy}{dx} = x + y using Euler's method with a step size of 0.1:Step 1: Given the initial condition y(0)=1 y(0) = 1 , the first approximation is y1=y0+hf(x0,y0) y_1 = y_0 + h f(x_0, y_0) , where h h is the step size, x0=0 x_0 = 0 , and y0=1 y_0 = 1 : y1=1+0.1(0+1)=1.1. y_1 = 1 + 0.1(0 + 1) = 1.1. Step 2: For the next approximation, x1=x0+h=0+0.1=0.1 x_1 = x_0 + h = 0 + 0.1 = 0.1 , and y1=1.1 y_1 = 1.1 , so: y2=1.1+0.1(0.1+1.1)=1.22. y_2 = 1.1 + 0.1(0.1 + 1.1) = 1.22. Step 3: Continue the process until reaching x=0.5 x = 0.5 . Step 4: Conclusion. The approximation of y y at x=0.5 x = 0.5 using Euler's method with a step size of 0.1 is approximately 1.22 1.22 . Solved on Solvelet with Basic AI Model
Some of the related questions asked by William Green on Solvelet
1. Use Euler's method with a step size of 0.1 0.1 to approximate the solution of the initial value problem y′=−2y+2 y' = -2y + 2 with y(0)=1 y(0) = 1 on the interval [0,1] [0, 1] .2. Approximate the solution of the initial value problem y′=3y−2 y' = 3y - 2 with y(0)=0 y(0) = 0 using Euler's method with a step size of 0.2 0.2 on the interval [0,1] [0, 1] .
DefinitionEuler’s method is a numerical integration method for ODEs. The idea of this method is to approximate the value of the solution at discrete points along the domain. At each point the method relies on the idea of the pointwise-derivative provided by the tangent-line, using this to guess the next value of the solution. Euler’s method is easy to implement, however it may be erroneous due to its first-order accuracy and the potentially rigid dependence on step size. It is widely used across physics, engineering, and scientific computing disciplines, within the field of numerical analysis for solving initial value problems of ordinary differential equations and simulating dynamical systems. Some Example Problems Eulers Method dxdy​=x2+y with y0​=1 [0,1]
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