Euler Method
New y = old y + (step size)(slope at old point)
The Euler Method, named after the Swiss mathematician Leonhard Euler, is a numerical method used to solve differential equations. It is a first-order numerical method that approximates the solution of a differential equation by updating the solution at each step by a linear approximation.
The Euler method starts with an initial value problem, where we have an ordinary differential equation and an initial value. The goal is to approximate the solution at each time step from the previous step.
The method computes the approximate solution by using the following algorithm:
1. Start with an initial value y0 at t0.
2. Choose a step size, h.
3. Compute yn+1 = yn + hf(tn, yn) at each time tn = t0 + nh, n = 0, 1, 2, 3, …
4. Repeat step 3 until the desired solution is reached.
Here, f(t, y) is the differential equation we want to solve. The value of yn+1 is obtained by taking the previous value yn, adding the product of the step size h and the slope of the tangent line at (tn, yn) which is given by f(tn, yn).
The Euler method has limitations, primarily due to its first-order accuracy. Thus, it is not the most precise method but is useful when more complex methods are not feasible or when a quick estimate is necessary. In practice, it can be improved by using smaller time steps or by using higher-order methods that utilize multiple points to estimate the solution.
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