First Order Differential Equations:
Consider the first order differential equation;
subject to y(x = x0) = y0. On integrating the above equation with respect to x between the limits (x0, x), we get
In Picard’s method, to obtain nth successive approximation, we write above equations as
To terminate the process, we compare the values yn and yn−1 up to desired accuracy.
Simultaneous First Order Differential Equations:
Consider the simultaneous first order differential equation;
Piacrd’s successive approximation for these equation is;
Using these equations, we may iterate the process up to desired accuracy of y and z.
Second Order Differential Equations:
Consider the second order differential equation;
By putting dy/dz = z, the above equation reduces to first order simultaneous differential equations:
Now these equation can be solved as discussed in previous section.