## Euler's Method

**Euler's Method:**

**First Order Differential Equations:**

Consider the differential equation;

where y(x_{0}) = y_{0}.

Suppose that we wish to find successively y_{1}, y_{2}, ..., y_{n} where y_{m} is the value of y corresponding to x = x_{m}, where xm = x_{0} mh, m = 1, 2, ...,h being small. Here, we use the property that in a small interval, a curve is nearly a straight line. Thus, in the interval x_{0} to x_{1} of x, we approximate the curve by the tangent at the point (x_{0}, y_{0}). Therefore, the equation of tangent at (x_{0}, y_{0}) is;

Hence, the value of y corresponding to x = x_{1} is

Since the curve is approximated by the tangent in [x_{0}, x_{1}], Equation above gives the approximated value of y_{1}.

**Simialrly,** approximating the curve int he next interval [x_{1}, x_{2}] by a line through (x_{1}, y_{1}) with slope f (x_{1}, y_{1}), we get;

Proceeding on, in general it can be shown that

**Simultaneous First Order Differential Equations:**

Euler’s methed can be extended to the solution of system of ordinary differential equations. The folowing example illustrates the procedure.

**Example:** using Euler’s method, assuming that x = 0, y = 4,and z = 6. Integrate to x = 2 with h = 0.5.

**Solution:** We have;

Euler’s formula is