Homogenous Equation - Example 2

Let us solve the differential equation

$x' = - x, \hspace{0.5cm} x(0) = 1. \hfill (25)$

The numerical solution of (25) by the Taylor series method is

$x_{n+1} = x_n*( 1 - h +\frac{ h^2}{2!} -\frac{ h^3}{3!} + \ldots). \hfill (26)$

Note : The exact solution of (25) is

$z = e^{-t}. \hfill (27)$

In combination with (20) the equation (25) can be called a "check function". The reason for this is that for the product of the analytical solution of equations (20) and (25) we have

$z= y * x = e^t * e^{-t} = 1 \hfill (28)$

and thus we can use (28) for testing the accuracy of the numerical solution.












The system of equations (20),(25),(28) has been selected in such a way that the high accuracy of the computation is again shown. The function z is constantly equal to 1 ( Fig.4.2).

 

Figure 4.2:

The computation in Fig.4.2 starts at time point zero and terminates at time t=1s. Corresponding values of the variables at time point t=1s are shown in the upper right-hand corner of the Fig.4.2.

Note:

Similarly, a check function of a homogenous equation

$y''+ y = 0\hspace{0.5cm} y(0)=0, \hspace{0.5cm} y'(0)=1$

(or equivalent system

$y'= x\hspace{0.5cm}y(0)=0,$

$x'=-y \hspace{0.5cm}x(0)=1)$

can be written in a form

z=x²+y²=1.

Time functions of x, y and ERR ( ERR=z-1) are in Fig.4.3.

 

Figure 4.3:

Conclusion: The solution of homogenous differential equations is one of the most important applications of the Modern Taylor Series Method.