The Euler’s Method Calculator is utilized to approximate solutions to differential equations by implementing the Euler's method, which is a first order numerical method to solve first degree differential equation with a given initial value.
“The first order numeric technique that is utilized to approximate the solutions of ordinary differential equations (ODE) of a given value is known as Euler's method”.
This method is taken name by the name of Swiss mathematician Euler. It involves estimating the values of unknown functions at discrete points and breaking down differential equations into smaller steps.
Euler’s method formula is a simple iterative approach that determines the first order ordinary differential equations. The formula essentially updates the values of x and y at each step, moving along the curve defined by the differential equation in discrete increments.
Use the following formulas iteratively to calculate the next approximations of x and y:
$$ x_{n+1} = x_n + h $$
$$ y_{n+1} = y_n + h . f(x_n,y_n) $$
Euler's Method proceeds as follows:
Our Euler’s method calculator starts by evaluating the slope of the tangent line to the solution curve at the initial point and after that uses this slope to determine the solution at the next point.
Let's say we have a simple function (x^2+4y)^(1/2) that needs to approximate the value of y at x = 2, given the initial condition y(0) = 1 with a step size (h) of 0.5
y(2) for $$ y' = \sqrt{x^{2} + 4 y} $$ when h = 0.5 and y(0) = 1
Now, we'll iteratively apply Euler's Method:
$$ y_{n+1} = y_n + h . f(x_n,y_n) $$
Now, we have
$$ h=0.5,x_0 = 0, y_0 = 1, f(x,y)=\sqrt{x^{2} + 4 y} $$
$$ x_1 = x_0 + h = 0 + 0.5 = 0.5 $$
$$ y(x_1) = y(0.5) = y_1 = y_0 + h. f(x_0,y_0) = 1 + 0.5 \times f(0,1) $$
$$ = 1 + 0.5 \times 2.0 = 2.0 $$
$$ x_2 = x_1 + h = 0.5+0.5=1 $$
$$ y(x_2) = y(1) = y_2= y_1 + h. f(x_1,y_1) = 2.0 + 0.5 \times f(0.5,2.0) $$
$$ = 2.0 + 0.5 \times 2.8722813232690143 = 3.436140661634507 $$
$$ x_3 = x_2 + h = 1+0.5=1.5 $$
$$ y(x_3) = y(1.5) = y_3= y_2 + h. f(x_2,y_2) = 3.436140661634507 + 0.5 \times f(1,3.436140661634507) $$
$$ = 3.436140661634507 + 0.5 \times 3.8398649255589743 = 5.356073124413994 $$
$$ x_4 = x_3 + h = 1.5 + 0.5 = 2 $$
$$ y(x_4) = y(2) = y_4= y_3 + h. f(x_3,y_3) = 5.356073124413994 + 0.5 \times f(1.5,5.356073124413994) $$
$$ = 5.356073124413994 + 0.5 \times 4.865623546643942 = 7.788884897735965 $$
$$ y(2) = = 7.788884897735965 $$
The below table helps to make your Euler's method calculator more engaging and informative.
Step | x0 | y0 | slope | yn |
---|---|---|---|---|
1 | 0 | 1 | 2.0 | 2.0 |
2 | 0.5 | 2.0 | 2.8722813232690143 | 3.436140661634507 |
3 | 1 | 3.436140661634507 | 3.8398649255589743 | 5.356073124413994 |
4 | 1.5 | 5.356073124413994 | 4.865623546643942 | 7.788884897735965 |
This demonstrates how Euler's formula calculator efficiently approximates solutions to differential equations, making complex problems more manageable.
The Euler method calculator are typically used to solve simple differential equations and also to generate data for plotting the solution to differential equations. Solve differential equations with ease and precision by putting the below values.
Our online Euler's calculator with step size gives you the final results after inserting the above values in it.
Wikipedia: Euler method, Geometrical description, Derivation, Local truncation error, Global truncation error.
LibreTexts: Euler's Method, Examples Illustrating The Error in Euler’s Method, Truncation Error, Semilinear Equations and Variation of Parameters.
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