Let’s recall the very basic aspect of Differential Equation.
It can be expressed as,
Combining both the equations, we get,
Summarizing the equation as,
Thus, we arrive at, .
where is the step size (the smaller, the better), is right side of the differential equation.
Example 1: with the initial condition as .
Here, we have two approaches to find the solution of the Differential Equation, which are as follows:
1) Analytical Solution: This is the exact solution of an ordinary differential equation.
Integrating both sides,
2) Numerical Solution: This is just an approximation of the solution.
We require this formula,
along with initial conditions, and
Let’s continue with Example 1, to find the Numerical Solution.
Example 1: with the initial condition as ad step-size .
We know the formula, .
In this equation we put the values of at interval of because of Step-Size () and obtain the value of By initial conditions, we know and .
Here, we are assuming the slope to be same between two consecutive points of and , whereas in the exact case the slope is changing constantly.