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 \(∆x = x_1 − x_0\).