Enter the first-order differential equation, related values, and let this calculator solve it using Euler’s Method.
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Use this Euler’s method calculator to solve the first-order differential equation with the given initial condition using the Euler’s method. It also offers a step-by-step solution that shows how Euler's (iterative) procedure approximates the solution to the differential equation to find the next point on the solution curve.
“The Euler method is a first-order numerical method used to solve ordinary differential equations (ODEs) with specific initial values”
This method was invented by the Swiss mathematician Leonhard Euler. Basically, Euler's method makes use of the derivatives at a specific point to approximate the function's value at the next point. By using the tangent line, this estimates the solution of the differential equations.
Hence, it is important to consider that Euler’s method is a simplification of the iterative method and may not be well estimated. So using the smaller step size generally leads to more precise approximations.
y_{(n+1)} = y_{n} + h . f(x_{n}, y_{n})
Within an equation:
Using the Euler's method with a step size of 1 to approximate the value of x(4) for the initial value problem by having:
Step # 1 - Set Up Initial Values
Step # 2 - Use the Euler’s Method Formula
An Euler’s equation has different components - get the given values and find the missing ones. Once you have done this, put the values into the formula to approximate the solution of x (4).
Step # 3 - Perform Iterations
We will repeatedly apply the formula four times (n = 0, 1, 2, 3) to approximate x(4). For the user’s convenience, we have engaged these calculations in the form of the table below:
Iteration (n) |
t_{n} |
x_{n} |
f(t_{n}, x_{n}) |
x(_{n + 1}) |
0 |
0 |
1 |
f(0, 1) = 1 |
1 + 1 * 1 = 2 |
1 |
1 |
2 |
f(1, 2) = 2 |
2 + 1 * 2 = 4 |
2 |
2 |
4 |
f(2, 4) = 4 |
4 + 1 * 4 = 8 |
3 |
3 |
8 |
f(3, 8) = 8 |
8 + 1 * 8 = 16 |
Step # 4 - Interpretation
The approximated value of x(4) is 16. It is calculated by using Euler’s method with a step size of 1 and four iterations. This iteration procedure can be automated with the help of Euler’s method calculator considering the initial value for ODE.
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