Stability and Error Propagation of Euler Methods

Let us consider the initial-value differential equation in the linear form  

We assume that A is real and Jq is finite. The analytical solution of this differential equation is 

This solution is inherently stable for A 0. Under these conditions  

we examine the stability of the numerical solution of this problem obtained from using the explicit Euler method. Momentarily we ignore the truncation and roundoff errors. Applying Eq. (5.60), we obtain the recurrence equation 

Using the methods described in Sec, 3.6, we obtain the characteristic equation 

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