3.1: The Euler Method

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Although it is not always possible to find an analytical solution of (3.1) for $y = y(x)$, it is always possible to determine a unique numerical solution given an initial value $y(x_0) = y_0$, and provided $f(x, y)$ is a well-behaved function. The differential equation (3.1) gives us the slope $f(x_0, y_0)$ of the tangent line to the solution curve $y = y(x)$ at the point $(x_0, y_0)$. With a small step size $∆x = x_1 − x_0$, the initial condition $(x_0, y_0)$ can be marched forward to $(x_1, y_1)$ along the tangent line using Euler’s method (see Fig. $\PageIndex{1}$) $$y_1=y_0+\Delta xf(x_0, y_0).\nonumber$$

This solution $(x_1, y_1)$ then becomes the new initial condition and is marched forward to $(x_2, y_2)$ along a newly determined tangent line with slope given by $f(x_1, y_1)$. For small enough $∆x$, the numerical solution converges to the exact solution.

Figure $\PageIndex{1}$: The differential equation $dy/dx = f(x, y),$ $y(x_0) = y_0,$ is integrated to $x = x_1$ using the Euler method $y_1 = y_0 + ∆x f(x_0, y_0)$, with $∆x = x_1 − x_0$.