Ordinary differential equations: Slope field and solution curves
The forward Euler method
Forward Euler method The geometric approach to solving a first-order dynamical system of the form with initial value via a slope field, according to the motto 'go with the flow', leads to the forward Euler method for computing a numerical solution a GDV.
Let be a point on the solution curve.
Suppose that you want to know the solution at time for some time step .
If is small, you can approximate the graph of the function near by the tangent line .
Then you can approximate at the function value with .
This method can be repeated with time and as starting point.
Then: .
One more time for and as starting point.
Then: .
You can continue this approximation process as long as you wish:
The figure below shows how the approximations indeed differ from the actual solution curve, but for small steps the error will be limited.
In the figure below, the function values calculated for the initial value problem are shown in a line graph for the step size .

Formula is in this particular case equal to with . The exact solution of this initial value problem is .
As the step size becomes smaller in the Euler method, the graph of the calculated function values becomes smoother and is increasingly more resemblant to the graph of an exact solution.