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# Euler Method of Differential Equations | Runge–Kutta method | Derivation Of Euler's Formula | Differential Equations

Here's first and basic method of solving Differential Equations, Euler Method. Euler Method- Euler method (also called forward Euler method) is a first-order numerical procedure for solving ordinary differential equations 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.
Here's first and basic method of solving Differential Equations, Euler Method.

### Euler Method-

Euler method (also called forward Euler method) is a first-order numerical procedure for solving ordinary differential equations 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.

Sometimes there is no analytical solution to a first-order differential equation and a numerical solution must be sought. The first-order differential equation dy/dx = f(x, y) with initial condition y(x0) = y0 provides the slope f(x0, y0) of the tangent line to the solution curve y = y(x) at the point (x0, y0). With a small step size ∆x = x1 − x0, the initial condition (x0, y0) can be marched forward to (x1, y1) along the tangent line using Euler’s method (see figure):

y1 = y0 + ∆x f(x0, y0).

This solution (x1, y1) then becomes the new initial condition and is marched forward to (x2, y2) along a newly determined tangent line with slope given by f(x1, y1). For small enough ∆x, the numerical solution converges to the unique solution, when such a solution exists. There are better numerical methods than the Euler method, but the basic principle of marching the solution forward remains the same

Watch this video by Jeff Chasnov Sir and enjoy the concept:

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