Secular terms, asymptotic solutions

Compare the asymptotic solution obtained in part (a) with the exact solution for e = 0.1. Disoiss the diverging behavior of the asymptotic solution near x = 0. Does the asymptotic solution behave better near X = 0 as more terms are retained Observing the asymptotic solution, it shows that the second term is more singular (secular) than the first term, and the third term is more singular than the second term. Thus, it is seen that just like the cubic equations dealt with in Problems 6.5 and 6.6, where the cubic term is multiplied by a small parameter, this differential equation also suffers the same growth in singular behavior. 

The task is to find the asymptotic approximation of the solution as e- 0 uniformly over a long time interval 0main goal. The higher order corrections of 0(e) will be defined in order to identify both the secular terms and the time intervals of asymptotic fitness. 

This situation is standard for problems with a small parameter. The interval of fitness of the trivial asymptotic solution (4.1) cannot be too long. The small terms affect the solution for long times. On the long time scale t-lie one has to construct a new asymptotic solution depending on another typical time. A suitable new independent variable is t = et as one can guess by looking at the secular terms. 

Thus, there are secular terms in the asymptotic solution on the slow scale, which indicate that the approximation (4.4) is false for the long times T = 1/e. 

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